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Page 1
Pinnacle Day: 11th - 18th Trigonometry
Trigonometry
It is that branch of Mathematics which
deals with the measurements of the
sides and angles of a triangle and also
the problems associated with angles.
Important Formulae
In a right angled triangle, as shown below
sin = • ?
??
??
• ?????????? ? =
??
??
• ?????? ? =
??
??
• ?????? ? =
??
??
• ?????? ? =
??
??
• ?????? ? =
??
??
Note that
= , ?????????? ?
1
?????? ?
? ?????????? ? × ?????? ? = 1
Similarly, = ?????? ?
1
?????? ?
? ?????? ? × ?????? ? = 1
And, = ?????? ?
1
?????? ?
? ?????? ? × ?????? ? = 1
Also, ?????? ? =
?????? ?
?????? ?
? sin = cos (90° - ) ? ?
? cosec = sec (90° - ) ? ?
? tan = cot (90° - ) ? ?
Conversion of trigonometric ratios
? sin (-x) = - sin x
? cosec (-x) = - cosec x
? cos (-x) = cos x
? sec (-x) = sec x
? tan (- x) = - tan x
? cot (- x) = - cot x
Signs of the trigonometric ratios in
different quadrants :
x axis no change • ± ? ?
y axis changes like • ± ? ?
sin ? ? ?????? ?
tan ? ? ?????? ?
cosec ? ? ?????? ?
Important Identities
? ??????
2
? + ??????
2
? = 1
? ??????
2
? - ??????
2
? = 1
? ??????????
2
? - ??????
2
? = 1
We can derive all other identities from
these three basic identities.
Values of Trigonometric functions for
different values of . ?
The value of sin or cos never exceeds ? ?
1, whereas the value of sec or cosec is ? ?
always greater than or equal to 1.
Range of all trigonometric functions
Minimum Maximum
Sinx,cosx
(odd powers)
-1 + 1
Sinx,cosx
(even powers)
0 +1
Tanx,cotx
(odd powers)
- 8 + 8
Tanx,cotx
(even powers)
0 + 8
Secx,cosecx
(odd powers)
- 8 + 8
Secx,cosecx
(even powers)
+1 + 8
Maximum and Minimum value of the
trigonometric expressions of the
different form :
a sinx • ± ?? ????????
Maximum value = ??
2
+ ??
2
Minimum value = - ??
2
+ ??
2
a + b • ???? ??
2
?? ???? ??
2
??
If, a b Maximum value = a >
Minimum value = b
If a < b Maximum value = b
Minimum value = a
a + b • ???????? ??
2
?? ???? ??
2
??
Maximum value = 8
Minimum value = ( + ?? ?? )
2
a + b • ???? ??
2
?? ???????? ??
2
??
a + b • ???? ??
2
?? ???? ??
2
??
a + b • ???? ??
2
?? ???? ??
2
??
Maximum and Minimum value of these
three trigonometric expression is 8 ??????
2 respectively. ????
+ • ???? ??
2 ??
?? ???? ??
2 ??
??
Maximum value = 1
Minimum value = (use x = 45 ° )
Example : Find the minimum and
maximum value of + ???? ??
2
?? ???? ??
4
?? ?
Maximum value of + = 1 ???? ??
2
?? ???? ??
4
??
Minimum value = + = ???? ??
2
45° ???? ??
4
45°
+ =
1
2
1
4
3
4
sinx cosx • ( )
??
Maximum value = 8
Minimum value = (
1
2
)
??
Example : Find the minimum value of
? ???? ??
4
?? ???? ??
4
??
Minimum value = = (
1
2
)
4 1
16
Compound Formulae
? ?????? ?? + ?? ( )
= ?????? ?? . ?????? ?? + ?????? ?? . ?????? ??
? ?????? ?? - ?? ( )
= ?????? ?? . ?????? ?? - ?????? ?? . ?????? ??
? ?????? ?? + ?? ( )
= ???????? . ???????? - ???????? . ????????
? ?????? ?? - ?? ( )
= ???????? . ???????? + ???????? . ?????? ??
? sin 2A = 2 sin A cos A
=
2 ?????? ??
1 + ??????
2
??
? cos 2A = ??????
2
?? - ??????
2
??
= 1 - 2 ??????
2
??
? cos2A =
1 - ??????
2
??
1 + ??????
2
??
= 2 ??????
2
?? - 1
Tan 2A = •
2 ????????
1 - ??????
2
??
? sin 3A = 3 sinA – 4 ??????
3
??
? cos 3A = 4 ??????
3
?? - 3 ????????
? ?????? 3 ?? =
3 ?????? ?? – ??????
3
??
1 - 3 ??????
2
??
? ?????? ? . ?????? 2? . ?????? 4?
=
1
4
?????? 3?
? ?????? ? . ?????? 2? . ?????? 4?
=
1
4
?????? 3?
? ?????? ?. ?????? 2? . ?????? 4? = ?????? 3?
? Tan(A + B) =
???????? + ????????
1 - ( ???????? × ???????? )
? Tan(A - B) =
???????? - ????????
1 + ( ???????? × ???????? )
? Cot (A + B )=
???????? ???????? - 1
???????? + ????????
? Cot (A - B) =
???????? ???????? + 1
???????? - ????????
Transformations of sums or
differences into products
sin C + sin D = 2 sin cos •
?? + ??
2
?? - ??
2
sin C - sin D = 2 cos sin •
?? + ??
2
?? - ??
2
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Page 2
Pinnacle Day: 11th - 18th Trigonometry
Trigonometry
It is that branch of Mathematics which
deals with the measurements of the
sides and angles of a triangle and also
the problems associated with angles.
Important Formulae
In a right angled triangle, as shown below
sin = • ?
??
??
• ?????????? ? =
??
??
• ?????? ? =
??
??
• ?????? ? =
??
??
• ?????? ? =
??
??
• ?????? ? =
??
??
Note that
= , ?????????? ?
1
?????? ?
? ?????????? ? × ?????? ? = 1
Similarly, = ?????? ?
1
?????? ?
? ?????? ? × ?????? ? = 1
And, = ?????? ?
1
?????? ?
? ?????? ? × ?????? ? = 1
Also, ?????? ? =
?????? ?
?????? ?
? sin = cos (90° - ) ? ?
? cosec = sec (90° - ) ? ?
? tan = cot (90° - ) ? ?
Conversion of trigonometric ratios
? sin (-x) = - sin x
? cosec (-x) = - cosec x
? cos (-x) = cos x
? sec (-x) = sec x
? tan (- x) = - tan x
? cot (- x) = - cot x
Signs of the trigonometric ratios in
different quadrants :
x axis no change • ± ? ?
y axis changes like • ± ? ?
sin ? ? ?????? ?
tan ? ? ?????? ?
cosec ? ? ?????? ?
Important Identities
? ??????
2
? + ??????
2
? = 1
? ??????
2
? - ??????
2
? = 1
? ??????????
2
? - ??????
2
? = 1
We can derive all other identities from
these three basic identities.
Values of Trigonometric functions for
different values of . ?
The value of sin or cos never exceeds ? ?
1, whereas the value of sec or cosec is ? ?
always greater than or equal to 1.
Range of all trigonometric functions
Minimum Maximum
Sinx,cosx
(odd powers)
-1 + 1
Sinx,cosx
(even powers)
0 +1
Tanx,cotx
(odd powers)
- 8 + 8
Tanx,cotx
(even powers)
0 + 8
Secx,cosecx
(odd powers)
- 8 + 8
Secx,cosecx
(even powers)
+1 + 8
Maximum and Minimum value of the
trigonometric expressions of the
different form :
a sinx • ± ?? ????????
Maximum value = ??
2
+ ??
2
Minimum value = - ??
2
+ ??
2
a + b • ???? ??
2
?? ???? ??
2
??
If, a b Maximum value = a >
Minimum value = b
If a < b Maximum value = b
Minimum value = a
a + b • ???????? ??
2
?? ???? ??
2
??
Maximum value = 8
Minimum value = ( + ?? ?? )
2
a + b • ???? ??
2
?? ???????? ??
2
??
a + b • ???? ??
2
?? ???? ??
2
??
a + b • ???? ??
2
?? ???? ??
2
??
Maximum and Minimum value of these
three trigonometric expression is 8 ??????
2 respectively. ????
+ • ???? ??
2 ??
?? ???? ??
2 ??
??
Maximum value = 1
Minimum value = (use x = 45 ° )
Example : Find the minimum and
maximum value of + ???? ??
2
?? ???? ??
4
?? ?
Maximum value of + = 1 ???? ??
2
?? ???? ??
4
??
Minimum value = + = ???? ??
2
45° ???? ??
4
45°
+ =
1
2
1
4
3
4
sinx cosx • ( )
??
Maximum value = 8
Minimum value = (
1
2
)
??
Example : Find the minimum value of
? ???? ??
4
?? ???? ??
4
??
Minimum value = = (
1
2
)
4 1
16
Compound Formulae
? ?????? ?? + ?? ( )
= ?????? ?? . ?????? ?? + ?????? ?? . ?????? ??
? ?????? ?? - ?? ( )
= ?????? ?? . ?????? ?? - ?????? ?? . ?????? ??
? ?????? ?? + ?? ( )
= ???????? . ???????? - ???????? . ????????
? ?????? ?? - ?? ( )
= ???????? . ???????? + ???????? . ?????? ??
? sin 2A = 2 sin A cos A
=
2 ?????? ??
1 + ??????
2
??
? cos 2A = ??????
2
?? - ??????
2
??
= 1 - 2 ??????
2
??
? cos2A =
1 - ??????
2
??
1 + ??????
2
??
= 2 ??????
2
?? - 1
Tan 2A = •
2 ????????
1 - ??????
2
??
? sin 3A = 3 sinA – 4 ??????
3
??
? cos 3A = 4 ??????
3
?? - 3 ????????
? ?????? 3 ?? =
3 ?????? ?? – ??????
3
??
1 - 3 ??????
2
??
? ?????? ? . ?????? 2? . ?????? 4?
=
1
4
?????? 3?
? ?????? ? . ?????? 2? . ?????? 4?
=
1
4
?????? 3?
? ?????? ?. ?????? 2? . ?????? 4? = ?????? 3?
? Tan(A + B) =
???????? + ????????
1 - ( ???????? × ???????? )
? Tan(A - B) =
???????? - ????????
1 + ( ???????? × ???????? )
? Cot (A + B )=
???????? ???????? - 1
???????? + ????????
? Cot (A - B) =
???????? ???????? + 1
???????? - ????????
Transformations of sums or
differences into products
sin C + sin D = 2 sin cos •
?? + ??
2
?? - ??
2
sin C - sin D = 2 cos sin •
?? + ??
2
?? - ??
2
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Pinnacle Day: 11th - 18th Trigonometry
cos C + cos D = 2 cos •
?? + ??
2
??????
?? - ??
2
cos C - cos D = - 2 sin sin •
?? + ??
2
?? - ??
2
Concepts
? a sinX + b cosX = m
a cosX - b sinX = n
Then, + = + ??
2
??
2
??
2
??
2
Example : if 3 sin Find ? + 5 ?????? ? = 5 .
the value of 3cos - 5sin ? ? ?
3cos - 5sin ? ? = 5
2
+ 3
2
- 5
2
=± 3
? a secX + b tanX = m
a tanX - b secX = n
Then, - = - ??
2
??
2
??
2
??
2
If secx and tanx are replaced by cosecx
and cotx respectively . Then the outcome
will be the same.
Example : if 29 sec ? - 21 ?????? ? = 20
Find the value of 29 tan ? ? - 21 ?????? ?
Sol:-- = 29 ?????? ? - 21 ?????? ?
= 0 = 2 0
2
- 2 9
2
+ 2 1
2
? sec x + tan x =
1
?????? ?? - ?????? ??
? cosec x + cot x =
1
?????????? ?? - ?????? ??
Some Important results
(i). If A + B + C = 180 . then, °
? tanA + tanB + tanC = tanA tanB tan C
? cotA cotB + cotB cotC + cotC cotA = 1
(ii). If A + B + C = 90 . then, °
? tanA tanB + tanB tanC + tanC tanA = 1
? cotA + cotB + cotC = cotA cotB cotC
(iii). If A + B = 45 . then °
? (1 + tanA) (1 + tanB) = 2
? (1 - cotA) (1 - cotB) = 2
Variety Questions
Q.1. If , then ?nd ??????
4
?? + ?????? ² ?? = 1
the value of ??????
4
?? - 3 ?????? ² ?? .
SSC CHSL Tier II 02/11/2023
(a) 1 (b) 2 (c) 2 (d) 1 - -
Q.2 If cot
2
? = 1 - e
2
, then the value of
cosec ? + cot
3
? sec ? is:
SSC CPO 05/10/2023 (3rd Shift)
(a) (b) ( 2 - ??
2
)
3
2
( 1 - ??
2
)
(c) (d) ( 1 - ??
2
)
3
2
( 2 - ??
2
)
1
2
Q.3. If cosecA + cotA = a , then ?nd ??
the value of .
( ??
2
?? - 1 )
( ??
2
?? + 1 )
SSC CHSL 17/08/2023 (2nd Shift)
(a) cosA (b) tanA (c) (d)
1
????????
1
????????
Q.4. If and ?????? ? + ?????? ? = ?? ?????? ? -
, then which of the relation ?????? ? = ??
satis?es for given values of p and q ?
SSC CHSL 11/08/2023 (4th Shift)
(a) (b) ??
2
- ??
2
= 4 ???? ??
2
- ??
2
= 8 ????
(c) (d) ??
2
+ ??
2
= 4 ???? ??
2
+ ??
2
= 4 ????
Q.5. If is an acute angle and sin cos ? ? ?
= 2 cos
3
- cos , then the value of sin ?
1
4
?
is: ?
SSC CHSL 10/08/2023 (2nd Shift)
(a) (b)
15 - 1
8
15 - 1
4
(c) (d)
15 + 1
4
15 - 1
2
Q.6. Simplify the following: 7(cot 11° cot
13° cot 19° cot 29° cot 61°cot 71° cot 77°
cot 79°)
SSC CHSL 09/08/2023 (3rd Shift)
(a) 7 (b) - 7 (c) - 1 (d) 1
Q.7. The given expression is equal to :
??????
4
?? + ??????
4
??
1 - 2 ??????
2
?? ??????
2
??
SSC CHSL 09/08/2023 (2nd Shift)
(a) 1 (b) -1 (c) 0 (d) 2
Q.8. If cot A + cos A = p and cot A - cos A
= q, then which of the following relation
is correct?
SSC CHSL 03/08/2023 (4th Shift)
(a) = p
2
+ q
2
(b) = p² + q
2
1
16 ????
1
4 ????
(c) = p² - q² (d) 4pq = p² + q² 16 ????
Q.9. The value of the expression 4( ??????
6
A + A) - 6 ( A + A) + 8 is : ??????
6
??????
4
??????
4
SSC CHSL 02/08/2023 (4th Shift)
(a) 4 (b) 8 (c) 7 (d) 6
Q.10. Evaluate the following:
2 + 2 + 2 + 2 ?????? 8 ?
SSC CGL 27/07/2023 (3rd shift)
(a) 2cos (b) 2cos2 (c) sin2 (d) cos2 ? ? ? ?
Q.11. Find the value of
1 - ?????? 3 ?
1 + ?????? 3 ?
SSC CGL 21/07/2023 (2nd shift)
(a) sec 3 - tan 3 (b) (sec 3 - tan 3 )
3
? ? ? ?
(c) (sec 3 - tan 3 )
2
(d) sec 3 + tan 3 ? ? ? ?
Q.12. The solution of the equation cos
2
?
+ sin + 1 = 0 lies between : ?
Graduate Level 30/06/2023 (shift 1)
(a) ( , ) (b) ( , )
3 p
4
5 p
4
5 p
4
7 p
4
(c) ( , ) (d) (- , )
p
4
3 p
4
p
4
p
4
Q.13. What is the value of (1+ cotA +
tanA) (sinA - cosA) ( ) ?
???????? ????????
?????? ³ ?? - ?????? ³ ??
SSC CHSL Tier II (26/06/2023)
(a) 1 (b) -1 (c) 0 (d) 2
Q.14. If sin m + sin n = p, cos m + cos n =
q, then ?nd the value of sin m sin n + ×
cos m cos n. ×
SSC CHSL 20/03/2023 (2nd Shift)
(a) p
2
+ q
2
- 2 (b)
( ??
2
+ ??
2
- 2 )
2
(c) p + q - pq (d) p + q + pq
Q.15. If a cot + b cosec = p and b cot ? ?
+ a cosec = q then p² - q
2
is equal to ? ?
_____.
SSC CGL Tier II (07/03/2023)
(a) – (b) – (c) b – a (d) + ??
2
??
2
??
2
??
2
??
2
??
2
Q.16. If A is an acute angle, the
simpli?ed form of
?????? ( p - ?? ). ?????? (
??
2
+ ?? ) ?????? (- ?? )
?????? ( p + ? ) ?????? (
3??
2
+ ? ) ?????? ( 2p - ?? )
SSC CGL 13/12/2022 (4th Shift)
(a) Cos²A (b) Sin A (c) Sin²A (d) Cos A
Q.17. What is the value of the expression
cos 2A cos 2B + sin
2
(A - B) - sin²(A + B) ?
SSC CGL 12/12/2022 (1st Shift)
(a) sin (2A - 2B) (b) sin (2A + 2B)
(c) cos (2A + 2B) (d) cos (2A - 2B)
Q.18. If = sin
2
60° +
?? - ?? ??????
2
30°
1 + ??????
2
30°
4tan
2
45° - cosec² 60°, then the value of k
(correct to two decimal places) is:
SSC CGL 03/12/2022 (3rd Shift)
(a) 5.55 (b) - 6.83 (c) - 5.58 (d) 6.83
Q.19. If Cos ? + Sin ? = Cos ?, ?nd the 2
value of (cos ? - Sin?).
SSC CPO 10/11/2022 (Afternoon)
(a) Sin ? (b) Cos ? 2 2
(c) Sin ? (d) Cos ?
1
2
1
2
Q.20. The value of is: ?????? 73° + ?????? 137°
SSC CHSL 10/06/2022 (Evening)
(a) sin 13 (b) cos 13 ° °
(c) cos 18 (d) sin 18 ° °
Q.21. The value of sin18° is given as
. Using this value, ?nd the value
5 - 1
4
of cos18°.
SSC CHSL 30/05/2022 (Afternoon)
(a) (b)
10 + 2 5
4
5 + 1
4
(c) (d)
6 + 2
4
10 - 2 5
4
Q.22. The value of is:
?????? 4 ? + ?????? 2 ?
?????? 4 ? + ?????? 2 ?
SSC CHSL 27/05/2022 ( Morning )
(a) (b) (c) (d) ?????? 3 ? ?????? 3 ? ?????? ? ?????? ?
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Page 3
Pinnacle Day: 11th - 18th Trigonometry
Trigonometry
It is that branch of Mathematics which
deals with the measurements of the
sides and angles of a triangle and also
the problems associated with angles.
Important Formulae
In a right angled triangle, as shown below
sin = • ?
??
??
• ?????????? ? =
??
??
• ?????? ? =
??
??
• ?????? ? =
??
??
• ?????? ? =
??
??
• ?????? ? =
??
??
Note that
= , ?????????? ?
1
?????? ?
? ?????????? ? × ?????? ? = 1
Similarly, = ?????? ?
1
?????? ?
? ?????? ? × ?????? ? = 1
And, = ?????? ?
1
?????? ?
? ?????? ? × ?????? ? = 1
Also, ?????? ? =
?????? ?
?????? ?
? sin = cos (90° - ) ? ?
? cosec = sec (90° - ) ? ?
? tan = cot (90° - ) ? ?
Conversion of trigonometric ratios
? sin (-x) = - sin x
? cosec (-x) = - cosec x
? cos (-x) = cos x
? sec (-x) = sec x
? tan (- x) = - tan x
? cot (- x) = - cot x
Signs of the trigonometric ratios in
different quadrants :
x axis no change • ± ? ?
y axis changes like • ± ? ?
sin ? ? ?????? ?
tan ? ? ?????? ?
cosec ? ? ?????? ?
Important Identities
? ??????
2
? + ??????
2
? = 1
? ??????
2
? - ??????
2
? = 1
? ??????????
2
? - ??????
2
? = 1
We can derive all other identities from
these three basic identities.
Values of Trigonometric functions for
different values of . ?
The value of sin or cos never exceeds ? ?
1, whereas the value of sec or cosec is ? ?
always greater than or equal to 1.
Range of all trigonometric functions
Minimum Maximum
Sinx,cosx
(odd powers)
-1 + 1
Sinx,cosx
(even powers)
0 +1
Tanx,cotx
(odd powers)
- 8 + 8
Tanx,cotx
(even powers)
0 + 8
Secx,cosecx
(odd powers)
- 8 + 8
Secx,cosecx
(even powers)
+1 + 8
Maximum and Minimum value of the
trigonometric expressions of the
different form :
a sinx • ± ?? ????????
Maximum value = ??
2
+ ??
2
Minimum value = - ??
2
+ ??
2
a + b • ???? ??
2
?? ???? ??
2
??
If, a b Maximum value = a >
Minimum value = b
If a < b Maximum value = b
Minimum value = a
a + b • ???????? ??
2
?? ???? ??
2
??
Maximum value = 8
Minimum value = ( + ?? ?? )
2
a + b • ???? ??
2
?? ???????? ??
2
??
a + b • ???? ??
2
?? ???? ??
2
??
a + b • ???? ??
2
?? ???? ??
2
??
Maximum and Minimum value of these
three trigonometric expression is 8 ??????
2 respectively. ????
+ • ???? ??
2 ??
?? ???? ??
2 ??
??
Maximum value = 1
Minimum value = (use x = 45 ° )
Example : Find the minimum and
maximum value of + ???? ??
2
?? ???? ??
4
?? ?
Maximum value of + = 1 ???? ??
2
?? ???? ??
4
??
Minimum value = + = ???? ??
2
45° ???? ??
4
45°
+ =
1
2
1
4
3
4
sinx cosx • ( )
??
Maximum value = 8
Minimum value = (
1
2
)
??
Example : Find the minimum value of
? ???? ??
4
?? ???? ??
4
??
Minimum value = = (
1
2
)
4 1
16
Compound Formulae
? ?????? ?? + ?? ( )
= ?????? ?? . ?????? ?? + ?????? ?? . ?????? ??
? ?????? ?? - ?? ( )
= ?????? ?? . ?????? ?? - ?????? ?? . ?????? ??
? ?????? ?? + ?? ( )
= ???????? . ???????? - ???????? . ????????
? ?????? ?? - ?? ( )
= ???????? . ???????? + ???????? . ?????? ??
? sin 2A = 2 sin A cos A
=
2 ?????? ??
1 + ??????
2
??
? cos 2A = ??????
2
?? - ??????
2
??
= 1 - 2 ??????
2
??
? cos2A =
1 - ??????
2
??
1 + ??????
2
??
= 2 ??????
2
?? - 1
Tan 2A = •
2 ????????
1 - ??????
2
??
? sin 3A = 3 sinA – 4 ??????
3
??
? cos 3A = 4 ??????
3
?? - 3 ????????
? ?????? 3 ?? =
3 ?????? ?? – ??????
3
??
1 - 3 ??????
2
??
? ?????? ? . ?????? 2? . ?????? 4?
=
1
4
?????? 3?
? ?????? ? . ?????? 2? . ?????? 4?
=
1
4
?????? 3?
? ?????? ?. ?????? 2? . ?????? 4? = ?????? 3?
? Tan(A + B) =
???????? + ????????
1 - ( ???????? × ???????? )
? Tan(A - B) =
???????? - ????????
1 + ( ???????? × ???????? )
? Cot (A + B )=
???????? ???????? - 1
???????? + ????????
? Cot (A - B) =
???????? ???????? + 1
???????? - ????????
Transformations of sums or
differences into products
sin C + sin D = 2 sin cos •
?? + ??
2
?? - ??
2
sin C - sin D = 2 cos sin •
?? + ??
2
?? - ??
2
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Pinnacle Day: 11th - 18th Trigonometry
cos C + cos D = 2 cos •
?? + ??
2
??????
?? - ??
2
cos C - cos D = - 2 sin sin •
?? + ??
2
?? - ??
2
Concepts
? a sinX + b cosX = m
a cosX - b sinX = n
Then, + = + ??
2
??
2
??
2
??
2
Example : if 3 sin Find ? + 5 ?????? ? = 5 .
the value of 3cos - 5sin ? ? ?
3cos - 5sin ? ? = 5
2
+ 3
2
- 5
2
=± 3
? a secX + b tanX = m
a tanX - b secX = n
Then, - = - ??
2
??
2
??
2
??
2
If secx and tanx are replaced by cosecx
and cotx respectively . Then the outcome
will be the same.
Example : if 29 sec ? - 21 ?????? ? = 20
Find the value of 29 tan ? ? - 21 ?????? ?
Sol:-- = 29 ?????? ? - 21 ?????? ?
= 0 = 2 0
2
- 2 9
2
+ 2 1
2
? sec x + tan x =
1
?????? ?? - ?????? ??
? cosec x + cot x =
1
?????????? ?? - ?????? ??
Some Important results
(i). If A + B + C = 180 . then, °
? tanA + tanB + tanC = tanA tanB tan C
? cotA cotB + cotB cotC + cotC cotA = 1
(ii). If A + B + C = 90 . then, °
? tanA tanB + tanB tanC + tanC tanA = 1
? cotA + cotB + cotC = cotA cotB cotC
(iii). If A + B = 45 . then °
? (1 + tanA) (1 + tanB) = 2
? (1 - cotA) (1 - cotB) = 2
Variety Questions
Q.1. If , then ?nd ??????
4
?? + ?????? ² ?? = 1
the value of ??????
4
?? - 3 ?????? ² ?? .
SSC CHSL Tier II 02/11/2023
(a) 1 (b) 2 (c) 2 (d) 1 - -
Q.2 If cot
2
? = 1 - e
2
, then the value of
cosec ? + cot
3
? sec ? is:
SSC CPO 05/10/2023 (3rd Shift)
(a) (b) ( 2 - ??
2
)
3
2
( 1 - ??
2
)
(c) (d) ( 1 - ??
2
)
3
2
( 2 - ??
2
)
1
2
Q.3. If cosecA + cotA = a , then ?nd ??
the value of .
( ??
2
?? - 1 )
( ??
2
?? + 1 )
SSC CHSL 17/08/2023 (2nd Shift)
(a) cosA (b) tanA (c) (d)
1
????????
1
????????
Q.4. If and ?????? ? + ?????? ? = ?? ?????? ? -
, then which of the relation ?????? ? = ??
satis?es for given values of p and q ?
SSC CHSL 11/08/2023 (4th Shift)
(a) (b) ??
2
- ??
2
= 4 ???? ??
2
- ??
2
= 8 ????
(c) (d) ??
2
+ ??
2
= 4 ???? ??
2
+ ??
2
= 4 ????
Q.5. If is an acute angle and sin cos ? ? ?
= 2 cos
3
- cos , then the value of sin ?
1
4
?
is: ?
SSC CHSL 10/08/2023 (2nd Shift)
(a) (b)
15 - 1
8
15 - 1
4
(c) (d)
15 + 1
4
15 - 1
2
Q.6. Simplify the following: 7(cot 11° cot
13° cot 19° cot 29° cot 61°cot 71° cot 77°
cot 79°)
SSC CHSL 09/08/2023 (3rd Shift)
(a) 7 (b) - 7 (c) - 1 (d) 1
Q.7. The given expression is equal to :
??????
4
?? + ??????
4
??
1 - 2 ??????
2
?? ??????
2
??
SSC CHSL 09/08/2023 (2nd Shift)
(a) 1 (b) -1 (c) 0 (d) 2
Q.8. If cot A + cos A = p and cot A - cos A
= q, then which of the following relation
is correct?
SSC CHSL 03/08/2023 (4th Shift)
(a) = p
2
+ q
2
(b) = p² + q
2
1
16 ????
1
4 ????
(c) = p² - q² (d) 4pq = p² + q² 16 ????
Q.9. The value of the expression 4( ??????
6
A + A) - 6 ( A + A) + 8 is : ??????
6
??????
4
??????
4
SSC CHSL 02/08/2023 (4th Shift)
(a) 4 (b) 8 (c) 7 (d) 6
Q.10. Evaluate the following:
2 + 2 + 2 + 2 ?????? 8 ?
SSC CGL 27/07/2023 (3rd shift)
(a) 2cos (b) 2cos2 (c) sin2 (d) cos2 ? ? ? ?
Q.11. Find the value of
1 - ?????? 3 ?
1 + ?????? 3 ?
SSC CGL 21/07/2023 (2nd shift)
(a) sec 3 - tan 3 (b) (sec 3 - tan 3 )
3
? ? ? ?
(c) (sec 3 - tan 3 )
2
(d) sec 3 + tan 3 ? ? ? ?
Q.12. The solution of the equation cos
2
?
+ sin + 1 = 0 lies between : ?
Graduate Level 30/06/2023 (shift 1)
(a) ( , ) (b) ( , )
3 p
4
5 p
4
5 p
4
7 p
4
(c) ( , ) (d) (- , )
p
4
3 p
4
p
4
p
4
Q.13. What is the value of (1+ cotA +
tanA) (sinA - cosA) ( ) ?
???????? ????????
?????? ³ ?? - ?????? ³ ??
SSC CHSL Tier II (26/06/2023)
(a) 1 (b) -1 (c) 0 (d) 2
Q.14. If sin m + sin n = p, cos m + cos n =
q, then ?nd the value of sin m sin n + ×
cos m cos n. ×
SSC CHSL 20/03/2023 (2nd Shift)
(a) p
2
+ q
2
- 2 (b)
( ??
2
+ ??
2
- 2 )
2
(c) p + q - pq (d) p + q + pq
Q.15. If a cot + b cosec = p and b cot ? ?
+ a cosec = q then p² - q
2
is equal to ? ?
_____.
SSC CGL Tier II (07/03/2023)
(a) – (b) – (c) b – a (d) + ??
2
??
2
??
2
??
2
??
2
??
2
Q.16. If A is an acute angle, the
simpli?ed form of
?????? ( p - ?? ). ?????? (
??
2
+ ?? ) ?????? (- ?? )
?????? ( p + ? ) ?????? (
3??
2
+ ? ) ?????? ( 2p - ?? )
SSC CGL 13/12/2022 (4th Shift)
(a) Cos²A (b) Sin A (c) Sin²A (d) Cos A
Q.17. What is the value of the expression
cos 2A cos 2B + sin
2
(A - B) - sin²(A + B) ?
SSC CGL 12/12/2022 (1st Shift)
(a) sin (2A - 2B) (b) sin (2A + 2B)
(c) cos (2A + 2B) (d) cos (2A - 2B)
Q.18. If = sin
2
60° +
?? - ?? ??????
2
30°
1 + ??????
2
30°
4tan
2
45° - cosec² 60°, then the value of k
(correct to two decimal places) is:
SSC CGL 03/12/2022 (3rd Shift)
(a) 5.55 (b) - 6.83 (c) - 5.58 (d) 6.83
Q.19. If Cos ? + Sin ? = Cos ?, ?nd the 2
value of (cos ? - Sin?).
SSC CPO 10/11/2022 (Afternoon)
(a) Sin ? (b) Cos ? 2 2
(c) Sin ? (d) Cos ?
1
2
1
2
Q.20. The value of is: ?????? 73° + ?????? 137°
SSC CHSL 10/06/2022 (Evening)
(a) sin 13 (b) cos 13 ° °
(c) cos 18 (d) sin 18 ° °
Q.21. The value of sin18° is given as
. Using this value, ?nd the value
5 - 1
4
of cos18°.
SSC CHSL 30/05/2022 (Afternoon)
(a) (b)
10 + 2 5
4
5 + 1
4
(c) (d)
6 + 2
4
10 - 2 5
4
Q.22. The value of is:
?????? 4 ? + ?????? 2 ?
?????? 4 ? + ?????? 2 ?
SSC CHSL 27/05/2022 ( Morning )
(a) (b) (c) (d) ?????? 3 ? ?????? 3 ? ?????? ? ?????? ?
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Pinnacle Day: 11th - 18th Trigonometry
Q.23. The value of - + ??????
2
46° ??????
2
44°
( ??????
2
1° + ??????
2
3° + ??????
2
5° +... + ??????
2
89° )
SSC CHSL 10/08/2021 (Evening)
(a) 23.5 (b) 20.5 (c) 22.5 (d) 21.5
Q.24. If A lies between 45° and 540°, and
sinA = 0.5, what is the value of in
??
3
degree ?
SSC CHSL 15/04/2021 (Morning)
(a) 170° (b) 175° (c) 165° (d) 160°
Q.25. If cos 27° = x, then the value of tan
63° is:
SSC CHSL 26/10/2020 (Evening)
(a) (b)
1 + ??
2
??
??
1 + ??
2
(c) (d)
1 - ??
2
??
??
1 - ??
2
Q.26. If ?? ??????
3
? + ?? ??????
3
? = ?????? ? ?????? ?
then the value of ?????? ?? ?????? ? = ?? ?????? ?,
is: ??
2
+ ??
2
SSC CHSL 20/10/2020 (Morning)
(a) 0 (b) 4 (c) 1 (d) 2
Q.27. If cos
2
? + cos
4
? = 1, then the value
of sin? + sin
2
? is:
SSC CHSL 16/10/2020 (Morning)
(a) 0 (b) (c) 2 (d) 1
1
2
Q.28. If sin? - cos? = , then ?nd the
7
17
value of sin? + cos?.
SSC CHSL 14/10/2020 (Evening)
(a) (b) (c) (d)
8
17
23
13
23
17
8
13
Q.29. If sec and ? ?????? ? ( 0° < ? < 90° )
are the roots of the equation
, then the value of k 6 ??
2
- ???? + 6 = 0
is:
SSC CHSL 14/10/2020 (Morning)
(a) (b) 3 (c) 2 (d) 3 3 2 3 3
Q.30. If cosA, sinA, cotA are in geometric
progression, then the value of tan
6
A -
tan
2
A is:
SSC CHSL 13/10/2020 (Afternoon)
(a) (b) 3 (c) (d) 1
1
2
1
3
Q.31. Find the value of Cos 225°
SSC CHSL 17/03/2020 (Afternoon)
(a) -0.7071 (b) -0.866
(c) 0.7071 (d) 0.866
Q.32. If 0° < A, B < 45°, cos(A + B) =
24
25
and sin(A - B) = , then tan2A = ?
15
17
SSC CGL 06/03/2020 (Evening)
(a) 0 (b) 1 (c) (d)
416
87
213
4
Q.33. If = then ??????
2
? - ??????
2
? ??????
2
?,
which of the following is true?
SSC CPO 13/12/2019 (Evening)
(a) cos cos = 1 ? ?
(b) cos cos = ? ? 2
(c) - = ??????
2
? ??????
2
? ??????
2
?
(d) - = ??????
2
? ??????
2
? ??????
2
?
Q.34 . everything between 0° and 90° For
it is given that a '
??
?? , ( ?? = 1 , 2 , 3 ......... 20 )
sin + sin + sin …………...sin = 20. a
1
a
2
a
3
a
20
Find the value (in degrees) of ( a
1
+
) . a
2
+ a
3
.... a
20
SSC CHSL 11/07/2019 (Morning)
(a) 1800° (b) 900° (c) 0° (d) 20°
Q.35. For and both being acute a ß
angles, it is given that sin( + ) = 1, cos a ß
( - ) = . The values of and are : a ß
1
2
a ß
SSC CHSL 10/07/2019 (Evening)
(a) (b) 75° , 15° 45° , 15°
(c) (d) 75° , 45° 60° , 30°
Q.36. If = , then what is the value of ? 9°
cot .cot .cot .cot .cot .cot .cot ? 2 ? 3 ? 4 ? 5 ? 6 ?
.cot .cot ? 7 ? 8 ? 9 ?
SSC CHSL 10/07/2019 (Evening)
(a) - 1 (b) 1 (c) (d) 3 3
1
3
Q.37. Which among the following is an
irrational quantity ?
SSC CHSL 10 /07/ 2019 (Afternoon)
(a) tan tan (b) sin 30° . 60° 30°
(c) tan (d) cos 45° 30°
Q.38. Which among the following
increases continuously in the range
? 0° ? ? ? 90°
SSC CHSL 09/07/2019 (Evening)
(a) cot (b) cosec (c) tan (d) cos ? ? ? ?
Q.39. If sec = 4 and tan = , ( ) ? ?? ?
4
??
?? ? 0
then the value of 8( ) is : ??
2
-
1
??
2
SSC CHSL 05/07/2019 (Afternoon)
(a) (b) (c) (d)
1
16
1
4
1
2
1
8
Q.40. If ×
1 - ?????? ?
1 + ?????? ?
?????????? ? - ?????? ?
?????????? ? + ?????? ?
= , then the value of r is:
1 - ??
1 + ??
SSC CHSL 03/07/2019 (Afternoon)
(a) Sin ?? (b) Cosec ?? (c) Sec ?? (d) Cos ??
Q.41. If =
( ?????? ? - ?????????? ?)( ?????? ? - ?????? ?)
??????
2
? - ??????
2
?
??
3
, then r = ?
SSC CHSL 03/07/2019 (Morning)
(a) Sin Cos (b) tan ? ? ?
(c) Cot (d) Cosec Sec ? ? ?
Q.42. If 3 - 2 - 3Cos ?? = 0, 0° < ?? < ??????
2
?
90° , then what is the value of (2Cosec ?? +
tan ?? ) ?
SSC CHSL 02/07/2019 (Evening)
(a) 7v3 (b) 5v3 (c) (d)
5 3
3
7 3
3
Q.43. A simpli?ed of
( + )( ) is :
?????? ?
1 + ?????? ?
1 + ?????? ?
?????? ?
1
?????? ? + ?????? ?
SSC CHSL 02/07/2019 (Evening)
(a) Cos (b) 2Sin (c) Sin (d) 2Cos ? ? ? ?
Q.44. The value of
is:
( ?????? 9° + ?????? 81° )( ?????? 9° + ?????????? 81° )
?????? 56° . ?????? 34° + ?????? 25° . ?????????? 65°
SSC CHSL 02/07/2019 (Evening)
(a) (b) 4 (c) 2 (d) 1
1
2
Q.45. is
( ?????? ? + ?????? ?) ( 1 - ?????? ?)
?????????? ?( 1 + ?????? ?) ( ?????????? ? - ?????? ?)
equal to:
SSC CHSL 02/07/2019 (Morning)
(a) Sin (b) Sec (c) Cos (d) Cosec ? ? ? ?
Q.46. If 3cos
2
A + 7sin
2
A = 4, then what is
the value of cot A, given that A is an
acute angle ?
SSC CGL 13/06/2019 (Evening)
(a) 1 (b) (c) (d) 3
3
2
1
3
Q.47. If Cosec = , then Sin + Cos ?
13
12
? ?
- tan is equal to : ?
SSC CGL 11/06/2019 (Afternoon)
(a) (b) (c) (d) -
91
65
139
65
71
65
71
65
Q.48. The value of ??????
2
38° + ??????
2
52° +
is equal to: ??????
2
30° - ??????
2
45°
SSC CGL 10/06/2019 (Afternoon)
(a) (b) (c) (d)
1
3
1
4
3
4
1
2
Q.49. If Sec4 = Cosec ( + 20°) , then ? ? ?
is equal to:
SSC CGL 10/06/2019 (Afternoon)
(a) 22° (b) 18° (c) 14° (d) 20°
Q.50. If 3 sin = 2 0° < ?? < 90°, then ? ??????
2
?,
the value of ( ??????
2
? + ??????
2
? - ???? ??????
2
?)
is :
SSC CGL 10/06/2019 (Morning)
(a) -2 (b) - (c) (d) 2
7
3
7
3
Q.51. If , then k = ?
?????? ? + ?????? ?
?????? ? - ?????? ?
=
?? + 1
?? - 1
SSC CGL 10/06/2019 (Morning)
(a) cosec (b) sec (c) cos (d) sin ? ? ? ?
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Page 4
Pinnacle Day: 11th - 18th Trigonometry
Trigonometry
It is that branch of Mathematics which
deals with the measurements of the
sides and angles of a triangle and also
the problems associated with angles.
Important Formulae
In a right angled triangle, as shown below
sin = • ?
??
??
• ?????????? ? =
??
??
• ?????? ? =
??
??
• ?????? ? =
??
??
• ?????? ? =
??
??
• ?????? ? =
??
??
Note that
= , ?????????? ?
1
?????? ?
? ?????????? ? × ?????? ? = 1
Similarly, = ?????? ?
1
?????? ?
? ?????? ? × ?????? ? = 1
And, = ?????? ?
1
?????? ?
? ?????? ? × ?????? ? = 1
Also, ?????? ? =
?????? ?
?????? ?
? sin = cos (90° - ) ? ?
? cosec = sec (90° - ) ? ?
? tan = cot (90° - ) ? ?
Conversion of trigonometric ratios
? sin (-x) = - sin x
? cosec (-x) = - cosec x
? cos (-x) = cos x
? sec (-x) = sec x
? tan (- x) = - tan x
? cot (- x) = - cot x
Signs of the trigonometric ratios in
different quadrants :
x axis no change • ± ? ?
y axis changes like • ± ? ?
sin ? ? ?????? ?
tan ? ? ?????? ?
cosec ? ? ?????? ?
Important Identities
? ??????
2
? + ??????
2
? = 1
? ??????
2
? - ??????
2
? = 1
? ??????????
2
? - ??????
2
? = 1
We can derive all other identities from
these three basic identities.
Values of Trigonometric functions for
different values of . ?
The value of sin or cos never exceeds ? ?
1, whereas the value of sec or cosec is ? ?
always greater than or equal to 1.
Range of all trigonometric functions
Minimum Maximum
Sinx,cosx
(odd powers)
-1 + 1
Sinx,cosx
(even powers)
0 +1
Tanx,cotx
(odd powers)
- 8 + 8
Tanx,cotx
(even powers)
0 + 8
Secx,cosecx
(odd powers)
- 8 + 8
Secx,cosecx
(even powers)
+1 + 8
Maximum and Minimum value of the
trigonometric expressions of the
different form :
a sinx • ± ?? ????????
Maximum value = ??
2
+ ??
2
Minimum value = - ??
2
+ ??
2
a + b • ???? ??
2
?? ???? ??
2
??
If, a b Maximum value = a >
Minimum value = b
If a < b Maximum value = b
Minimum value = a
a + b • ???????? ??
2
?? ???? ??
2
??
Maximum value = 8
Minimum value = ( + ?? ?? )
2
a + b • ???? ??
2
?? ???????? ??
2
??
a + b • ???? ??
2
?? ???? ??
2
??
a + b • ???? ??
2
?? ???? ??
2
??
Maximum and Minimum value of these
three trigonometric expression is 8 ??????
2 respectively. ????
+ • ???? ??
2 ??
?? ???? ??
2 ??
??
Maximum value = 1
Minimum value = (use x = 45 ° )
Example : Find the minimum and
maximum value of + ???? ??
2
?? ???? ??
4
?? ?
Maximum value of + = 1 ???? ??
2
?? ???? ??
4
??
Minimum value = + = ???? ??
2
45° ???? ??
4
45°
+ =
1
2
1
4
3
4
sinx cosx • ( )
??
Maximum value = 8
Minimum value = (
1
2
)
??
Example : Find the minimum value of
? ???? ??
4
?? ???? ??
4
??
Minimum value = = (
1
2
)
4 1
16
Compound Formulae
? ?????? ?? + ?? ( )
= ?????? ?? . ?????? ?? + ?????? ?? . ?????? ??
? ?????? ?? - ?? ( )
= ?????? ?? . ?????? ?? - ?????? ?? . ?????? ??
? ?????? ?? + ?? ( )
= ???????? . ???????? - ???????? . ????????
? ?????? ?? - ?? ( )
= ???????? . ???????? + ???????? . ?????? ??
? sin 2A = 2 sin A cos A
=
2 ?????? ??
1 + ??????
2
??
? cos 2A = ??????
2
?? - ??????
2
??
= 1 - 2 ??????
2
??
? cos2A =
1 - ??????
2
??
1 + ??????
2
??
= 2 ??????
2
?? - 1
Tan 2A = •
2 ????????
1 - ??????
2
??
? sin 3A = 3 sinA – 4 ??????
3
??
? cos 3A = 4 ??????
3
?? - 3 ????????
? ?????? 3 ?? =
3 ?????? ?? – ??????
3
??
1 - 3 ??????
2
??
? ?????? ? . ?????? 2? . ?????? 4?
=
1
4
?????? 3?
? ?????? ? . ?????? 2? . ?????? 4?
=
1
4
?????? 3?
? ?????? ?. ?????? 2? . ?????? 4? = ?????? 3?
? Tan(A + B) =
???????? + ????????
1 - ( ???????? × ???????? )
? Tan(A - B) =
???????? - ????????
1 + ( ???????? × ???????? )
? Cot (A + B )=
???????? ???????? - 1
???????? + ????????
? Cot (A - B) =
???????? ???????? + 1
???????? - ????????
Transformations of sums or
differences into products
sin C + sin D = 2 sin cos •
?? + ??
2
?? - ??
2
sin C - sin D = 2 cos sin •
?? + ??
2
?? - ??
2
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Pinnacle Day: 11th - 18th Trigonometry
cos C + cos D = 2 cos •
?? + ??
2
??????
?? - ??
2
cos C - cos D = - 2 sin sin •
?? + ??
2
?? - ??
2
Concepts
? a sinX + b cosX = m
a cosX - b sinX = n
Then, + = + ??
2
??
2
??
2
??
2
Example : if 3 sin Find ? + 5 ?????? ? = 5 .
the value of 3cos - 5sin ? ? ?
3cos - 5sin ? ? = 5
2
+ 3
2
- 5
2
=± 3
? a secX + b tanX = m
a tanX - b secX = n
Then, - = - ??
2
??
2
??
2
??
2
If secx and tanx are replaced by cosecx
and cotx respectively . Then the outcome
will be the same.
Example : if 29 sec ? - 21 ?????? ? = 20
Find the value of 29 tan ? ? - 21 ?????? ?
Sol:-- = 29 ?????? ? - 21 ?????? ?
= 0 = 2 0
2
- 2 9
2
+ 2 1
2
? sec x + tan x =
1
?????? ?? - ?????? ??
? cosec x + cot x =
1
?????????? ?? - ?????? ??
Some Important results
(i). If A + B + C = 180 . then, °
? tanA + tanB + tanC = tanA tanB tan C
? cotA cotB + cotB cotC + cotC cotA = 1
(ii). If A + B + C = 90 . then, °
? tanA tanB + tanB tanC + tanC tanA = 1
? cotA + cotB + cotC = cotA cotB cotC
(iii). If A + B = 45 . then °
? (1 + tanA) (1 + tanB) = 2
? (1 - cotA) (1 - cotB) = 2
Variety Questions
Q.1. If , then ?nd ??????
4
?? + ?????? ² ?? = 1
the value of ??????
4
?? - 3 ?????? ² ?? .
SSC CHSL Tier II 02/11/2023
(a) 1 (b) 2 (c) 2 (d) 1 - -
Q.2 If cot
2
? = 1 - e
2
, then the value of
cosec ? + cot
3
? sec ? is:
SSC CPO 05/10/2023 (3rd Shift)
(a) (b) ( 2 - ??
2
)
3
2
( 1 - ??
2
)
(c) (d) ( 1 - ??
2
)
3
2
( 2 - ??
2
)
1
2
Q.3. If cosecA + cotA = a , then ?nd ??
the value of .
( ??
2
?? - 1 )
( ??
2
?? + 1 )
SSC CHSL 17/08/2023 (2nd Shift)
(a) cosA (b) tanA (c) (d)
1
????????
1
????????
Q.4. If and ?????? ? + ?????? ? = ?? ?????? ? -
, then which of the relation ?????? ? = ??
satis?es for given values of p and q ?
SSC CHSL 11/08/2023 (4th Shift)
(a) (b) ??
2
- ??
2
= 4 ???? ??
2
- ??
2
= 8 ????
(c) (d) ??
2
+ ??
2
= 4 ???? ??
2
+ ??
2
= 4 ????
Q.5. If is an acute angle and sin cos ? ? ?
= 2 cos
3
- cos , then the value of sin ?
1
4
?
is: ?
SSC CHSL 10/08/2023 (2nd Shift)
(a) (b)
15 - 1
8
15 - 1
4
(c) (d)
15 + 1
4
15 - 1
2
Q.6. Simplify the following: 7(cot 11° cot
13° cot 19° cot 29° cot 61°cot 71° cot 77°
cot 79°)
SSC CHSL 09/08/2023 (3rd Shift)
(a) 7 (b) - 7 (c) - 1 (d) 1
Q.7. The given expression is equal to :
??????
4
?? + ??????
4
??
1 - 2 ??????
2
?? ??????
2
??
SSC CHSL 09/08/2023 (2nd Shift)
(a) 1 (b) -1 (c) 0 (d) 2
Q.8. If cot A + cos A = p and cot A - cos A
= q, then which of the following relation
is correct?
SSC CHSL 03/08/2023 (4th Shift)
(a) = p
2
+ q
2
(b) = p² + q
2
1
16 ????
1
4 ????
(c) = p² - q² (d) 4pq = p² + q² 16 ????
Q.9. The value of the expression 4( ??????
6
A + A) - 6 ( A + A) + 8 is : ??????
6
??????
4
??????
4
SSC CHSL 02/08/2023 (4th Shift)
(a) 4 (b) 8 (c) 7 (d) 6
Q.10. Evaluate the following:
2 + 2 + 2 + 2 ?????? 8 ?
SSC CGL 27/07/2023 (3rd shift)
(a) 2cos (b) 2cos2 (c) sin2 (d) cos2 ? ? ? ?
Q.11. Find the value of
1 - ?????? 3 ?
1 + ?????? 3 ?
SSC CGL 21/07/2023 (2nd shift)
(a) sec 3 - tan 3 (b) (sec 3 - tan 3 )
3
? ? ? ?
(c) (sec 3 - tan 3 )
2
(d) sec 3 + tan 3 ? ? ? ?
Q.12. The solution of the equation cos
2
?
+ sin + 1 = 0 lies between : ?
Graduate Level 30/06/2023 (shift 1)
(a) ( , ) (b) ( , )
3 p
4
5 p
4
5 p
4
7 p
4
(c) ( , ) (d) (- , )
p
4
3 p
4
p
4
p
4
Q.13. What is the value of (1+ cotA +
tanA) (sinA - cosA) ( ) ?
???????? ????????
?????? ³ ?? - ?????? ³ ??
SSC CHSL Tier II (26/06/2023)
(a) 1 (b) -1 (c) 0 (d) 2
Q.14. If sin m + sin n = p, cos m + cos n =
q, then ?nd the value of sin m sin n + ×
cos m cos n. ×
SSC CHSL 20/03/2023 (2nd Shift)
(a) p
2
+ q
2
- 2 (b)
( ??
2
+ ??
2
- 2 )
2
(c) p + q - pq (d) p + q + pq
Q.15. If a cot + b cosec = p and b cot ? ?
+ a cosec = q then p² - q
2
is equal to ? ?
_____.
SSC CGL Tier II (07/03/2023)
(a) – (b) – (c) b – a (d) + ??
2
??
2
??
2
??
2
??
2
??
2
Q.16. If A is an acute angle, the
simpli?ed form of
?????? ( p - ?? ). ?????? (
??
2
+ ?? ) ?????? (- ?? )
?????? ( p + ? ) ?????? (
3??
2
+ ? ) ?????? ( 2p - ?? )
SSC CGL 13/12/2022 (4th Shift)
(a) Cos²A (b) Sin A (c) Sin²A (d) Cos A
Q.17. What is the value of the expression
cos 2A cos 2B + sin
2
(A - B) - sin²(A + B) ?
SSC CGL 12/12/2022 (1st Shift)
(a) sin (2A - 2B) (b) sin (2A + 2B)
(c) cos (2A + 2B) (d) cos (2A - 2B)
Q.18. If = sin
2
60° +
?? - ?? ??????
2
30°
1 + ??????
2
30°
4tan
2
45° - cosec² 60°, then the value of k
(correct to two decimal places) is:
SSC CGL 03/12/2022 (3rd Shift)
(a) 5.55 (b) - 6.83 (c) - 5.58 (d) 6.83
Q.19. If Cos ? + Sin ? = Cos ?, ?nd the 2
value of (cos ? - Sin?).
SSC CPO 10/11/2022 (Afternoon)
(a) Sin ? (b) Cos ? 2 2
(c) Sin ? (d) Cos ?
1
2
1
2
Q.20. The value of is: ?????? 73° + ?????? 137°
SSC CHSL 10/06/2022 (Evening)
(a) sin 13 (b) cos 13 ° °
(c) cos 18 (d) sin 18 ° °
Q.21. The value of sin18° is given as
. Using this value, ?nd the value
5 - 1
4
of cos18°.
SSC CHSL 30/05/2022 (Afternoon)
(a) (b)
10 + 2 5
4
5 + 1
4
(c) (d)
6 + 2
4
10 - 2 5
4
Q.22. The value of is:
?????? 4 ? + ?????? 2 ?
?????? 4 ? + ?????? 2 ?
SSC CHSL 27/05/2022 ( Morning )
(a) (b) (c) (d) ?????? 3 ? ?????? 3 ? ?????? ? ?????? ?
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Pinnacle Day: 11th - 18th Trigonometry
Q.23. The value of - + ??????
2
46° ??????
2
44°
( ??????
2
1° + ??????
2
3° + ??????
2
5° +... + ??????
2
89° )
SSC CHSL 10/08/2021 (Evening)
(a) 23.5 (b) 20.5 (c) 22.5 (d) 21.5
Q.24. If A lies between 45° and 540°, and
sinA = 0.5, what is the value of in
??
3
degree ?
SSC CHSL 15/04/2021 (Morning)
(a) 170° (b) 175° (c) 165° (d) 160°
Q.25. If cos 27° = x, then the value of tan
63° is:
SSC CHSL 26/10/2020 (Evening)
(a) (b)
1 + ??
2
??
??
1 + ??
2
(c) (d)
1 - ??
2
??
??
1 - ??
2
Q.26. If ?? ??????
3
? + ?? ??????
3
? = ?????? ? ?????? ?
then the value of ?????? ?? ?????? ? = ?? ?????? ?,
is: ??
2
+ ??
2
SSC CHSL 20/10/2020 (Morning)
(a) 0 (b) 4 (c) 1 (d) 2
Q.27. If cos
2
? + cos
4
? = 1, then the value
of sin? + sin
2
? is:
SSC CHSL 16/10/2020 (Morning)
(a) 0 (b) (c) 2 (d) 1
1
2
Q.28. If sin? - cos? = , then ?nd the
7
17
value of sin? + cos?.
SSC CHSL 14/10/2020 (Evening)
(a) (b) (c) (d)
8
17
23
13
23
17
8
13
Q.29. If sec and ? ?????? ? ( 0° < ? < 90° )
are the roots of the equation
, then the value of k 6 ??
2
- ???? + 6 = 0
is:
SSC CHSL 14/10/2020 (Morning)
(a) (b) 3 (c) 2 (d) 3 3 2 3 3
Q.30. If cosA, sinA, cotA are in geometric
progression, then the value of tan
6
A -
tan
2
A is:
SSC CHSL 13/10/2020 (Afternoon)
(a) (b) 3 (c) (d) 1
1
2
1
3
Q.31. Find the value of Cos 225°
SSC CHSL 17/03/2020 (Afternoon)
(a) -0.7071 (b) -0.866
(c) 0.7071 (d) 0.866
Q.32. If 0° < A, B < 45°, cos(A + B) =
24
25
and sin(A - B) = , then tan2A = ?
15
17
SSC CGL 06/03/2020 (Evening)
(a) 0 (b) 1 (c) (d)
416
87
213
4
Q.33. If = then ??????
2
? - ??????
2
? ??????
2
?,
which of the following is true?
SSC CPO 13/12/2019 (Evening)
(a) cos cos = 1 ? ?
(b) cos cos = ? ? 2
(c) - = ??????
2
? ??????
2
? ??????
2
?
(d) - = ??????
2
? ??????
2
? ??????
2
?
Q.34 . everything between 0° and 90° For
it is given that a '
??
?? , ( ?? = 1 , 2 , 3 ......... 20 )
sin + sin + sin …………...sin = 20. a
1
a
2
a
3
a
20
Find the value (in degrees) of ( a
1
+
) . a
2
+ a
3
.... a
20
SSC CHSL 11/07/2019 (Morning)
(a) 1800° (b) 900° (c) 0° (d) 20°
Q.35. For and both being acute a ß
angles, it is given that sin( + ) = 1, cos a ß
( - ) = . The values of and are : a ß
1
2
a ß
SSC CHSL 10/07/2019 (Evening)
(a) (b) 75° , 15° 45° , 15°
(c) (d) 75° , 45° 60° , 30°
Q.36. If = , then what is the value of ? 9°
cot .cot .cot .cot .cot .cot .cot ? 2 ? 3 ? 4 ? 5 ? 6 ?
.cot .cot ? 7 ? 8 ? 9 ?
SSC CHSL 10/07/2019 (Evening)
(a) - 1 (b) 1 (c) (d) 3 3
1
3
Q.37. Which among the following is an
irrational quantity ?
SSC CHSL 10 /07/ 2019 (Afternoon)
(a) tan tan (b) sin 30° . 60° 30°
(c) tan (d) cos 45° 30°
Q.38. Which among the following
increases continuously in the range
? 0° ? ? ? 90°
SSC CHSL 09/07/2019 (Evening)
(a) cot (b) cosec (c) tan (d) cos ? ? ? ?
Q.39. If sec = 4 and tan = , ( ) ? ?? ?
4
??
?? ? 0
then the value of 8( ) is : ??
2
-
1
??
2
SSC CHSL 05/07/2019 (Afternoon)
(a) (b) (c) (d)
1
16
1
4
1
2
1
8
Q.40. If ×
1 - ?????? ?
1 + ?????? ?
?????????? ? - ?????? ?
?????????? ? + ?????? ?
= , then the value of r is:
1 - ??
1 + ??
SSC CHSL 03/07/2019 (Afternoon)
(a) Sin ?? (b) Cosec ?? (c) Sec ?? (d) Cos ??
Q.41. If =
( ?????? ? - ?????????? ?)( ?????? ? - ?????? ?)
??????
2
? - ??????
2
?
??
3
, then r = ?
SSC CHSL 03/07/2019 (Morning)
(a) Sin Cos (b) tan ? ? ?
(c) Cot (d) Cosec Sec ? ? ?
Q.42. If 3 - 2 - 3Cos ?? = 0, 0° < ?? < ??????
2
?
90° , then what is the value of (2Cosec ?? +
tan ?? ) ?
SSC CHSL 02/07/2019 (Evening)
(a) 7v3 (b) 5v3 (c) (d)
5 3
3
7 3
3
Q.43. A simpli?ed of
( + )( ) is :
?????? ?
1 + ?????? ?
1 + ?????? ?
?????? ?
1
?????? ? + ?????? ?
SSC CHSL 02/07/2019 (Evening)
(a) Cos (b) 2Sin (c) Sin (d) 2Cos ? ? ? ?
Q.44. The value of
is:
( ?????? 9° + ?????? 81° )( ?????? 9° + ?????????? 81° )
?????? 56° . ?????? 34° + ?????? 25° . ?????????? 65°
SSC CHSL 02/07/2019 (Evening)
(a) (b) 4 (c) 2 (d) 1
1
2
Q.45. is
( ?????? ? + ?????? ?) ( 1 - ?????? ?)
?????????? ?( 1 + ?????? ?) ( ?????????? ? - ?????? ?)
equal to:
SSC CHSL 02/07/2019 (Morning)
(a) Sin (b) Sec (c) Cos (d) Cosec ? ? ? ?
Q.46. If 3cos
2
A + 7sin
2
A = 4, then what is
the value of cot A, given that A is an
acute angle ?
SSC CGL 13/06/2019 (Evening)
(a) 1 (b) (c) (d) 3
3
2
1
3
Q.47. If Cosec = , then Sin + Cos ?
13
12
? ?
- tan is equal to : ?
SSC CGL 11/06/2019 (Afternoon)
(a) (b) (c) (d) -
91
65
139
65
71
65
71
65
Q.48. The value of ??????
2
38° + ??????
2
52° +
is equal to: ??????
2
30° - ??????
2
45°
SSC CGL 10/06/2019 (Afternoon)
(a) (b) (c) (d)
1
3
1
4
3
4
1
2
Q.49. If Sec4 = Cosec ( + 20°) , then ? ? ?
is equal to:
SSC CGL 10/06/2019 (Afternoon)
(a) 22° (b) 18° (c) 14° (d) 20°
Q.50. If 3 sin = 2 0° < ?? < 90°, then ? ??????
2
?,
the value of ( ??????
2
? + ??????
2
? - ???? ??????
2
?)
is :
SSC CGL 10/06/2019 (Morning)
(a) -2 (b) - (c) (d) 2
7
3
7
3
Q.51. If , then k = ?
?????? ? + ?????? ?
?????? ? - ?????? ?
=
?? + 1
?? - 1
SSC CGL 10/06/2019 (Morning)
(a) cosec (b) sec (c) cos (d) sin ? ? ? ?
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Pinnacle Day: 11th - 18th Trigonometry
Q.52. If = 2
1
?????????? ? - 1
+
1
?????????? ? + 1
sec?? , 0° < ?? < 90° , then the value of
(cot?? + cos ?? ) is:
SSC CGL 07/06/2019 (Evening)
(a) (b)
1 + 2
2
2 + 2
2
(c) (d)
2 + 3
2
1 + 2
Q.53. If sin = , 0° < < 90°, ?
??
??
2
+ ??
2
?
then the value of sec + tan is : ? ?
SSC CGL 07/06/2019 (Afternoon)
(a) (b)
??
2
+ ??
2
+ ??
??
??
2
+ ??
2
+ ??
2 ??
(c) (d)
??
2
+ ??
2
+ ??
2 ??
??
2
+ ??
2
+ ??
??
Q.54. If - 3cos + 2 = , 0° < < ??????
2
? ? ??????
2
? ?
90°, then the value of 2cosec + 4cot is : ? ?
SSC CGL 07/06/2019 (Afternoon)
(a) (b) (c) (d)
8 3
3
4 3
4
2 3 4 3
Q.55. The value of ,
1
???????? - ????????
-
1
????????
0° < x < 90°, is equal to :
SSC CGL 07/06/2019 (Morning)
(a) 2secx (b) tanx (c) 2cosx (d) cotx
Q.56. If - 3sec + 3 = 0 , 0° < x < ??????
2
? ?
90°, then the value of sin + cot is : ? ?
SSC CGL 07/06/2019 (Morning)
(a) (b) (c) (d)
5 3
6
2 3
5 3
3
3 3
Q.57. If (1 + ??????
2
?) + ( 1 + ( ??????
2
?)
- 1
)
= k, then = ? ??
SSC CGL 06/06/2019 (Evening)
(a) cosec .sec (b) cosec .cos ? ? ? ?
(c) sin .cos (d) sin .sec ? ? ? ?
Q.58. The value of is
1
?????? ?
-
??????
2
?
1 + ?????????? ?
SSC CGL 06/06/2019 (Evening)
(a) 0 (b) 1 (c) 2 (d) -1
Q.59. ( ) = ?
1
1 + ??????
2
?
+
1
1 + ??????????
2
?
SSC CGL 06/06/2019 (Evening)
(a) (b) 1 (c) (d) 2 ??????
2
? ??????????
2
?
Q.60. If = 4, 0° < ??
?????? ?
1 - ?????? ?
+
?????? ?
1 + ?????? ?
< 90° , then the value of (tan ?? + cosec?? )
is :
SSC CGL 06/06/2019 (Evening)
(a) (b) (c) (d)
5 2
2
5 3
3
4 3
3
5 2
3
Q.61. = ?
?????? ? - ?????? ? + 1
?????? ? + ?????? ? - 1
SSC CGL 06/06/2019 (Morning)
(a) sec sin (b) sec tan ? ? ? ?
(c) sec + tan (d) sec - tan ? ? ? ?
Q.62. The value of
is ??????
2
? + ??????????
2
? × ??????
2
? - ??????
2
?
equal to :
SSC CGL 06/06/2019 (Morning)
(a) (b) ?????????? ? ??????
2
? ?????? ? ??????
2
?
(c) sin (d) ? ??????
2
? ?????????? ? ??????
2
?
Q.63. If sec - tan = P , then cosec = ? ? ? ?
SSC CGL 04/06/ 2019 (Evening)
(a) (b)
2 ??
1 - ??
2
1 - ??
2
1 + ??
2
(c) (d)
??
2
+ 1
1 - ??
2
2 ??
1 + ??
2
Q.64. If and 0
0
< ? < 90
0
??????
2
?
then the value of ( = 3 ( ??????
2
? - ??????
2
?)
1
2
)
- 1
is : ?????? ? + ?????? ?
SSC CGL 04/06/2019 (Afternoon)
(a) (b) 2(2 - 3 + 2 3 )
(c) 2( (d) 3 - 1 ) 3 + 1
Q.65. is (
?????? ? - 2 ??????
3
?
2 ??????
3
? - ?????? ?
)
2
+ 1 , ? ? 45
0
equal to.
SSC CGL 04/06/2019 (Afternoon)
(a) (b) (c) (d) 2 ??????????
2
? ??????
2
? ??????
2
? ??????
2
?
Q.66. (cosecA - sinA + (secA - cosA - )
2
)
2
(cotA - tanA is equal to: )
2
SSC CPO 16/03/2019 (Evening)
(a) 2 (b) 0 (c) 1 (d) -1
Q.67. Find the value of 1 +
??????
2
??
1 + ?????? ??
SSC CPO 15/03/2019 (Evening)
(a) cosec A (b) cos A (c) sec A (d) sin A
Q.68. If x = a cos + b sin and y = a sin - ? ?
b cos , the value of + is: ? ??
2
??
2
SSC CPO 15/03/2019 (Morning)
(a) - (b) a - b (c) + (d) a + b ??
2
??
2
??
2
??
2
Practice Questions
SSC CPO 2023 Tier - 1
Q.69. Find the value of (tan
2
? + tan
4
?).
SSC CPO 03/10/2023 (1st Shift)
(a) (b) ??????
2
? - ??????
4
? ??????
2
? - ??????
2
?
(c) (d) ??????
4
? + ??????
2
? ??????
4
? - ??????
2
?
Q.70. If ? is an acute angle and sin ? +
cosec ? = 2,then the value of ? + ??????
5
? is: ??????????
5
SSC CPO 03/10/2023 (1st Shift)
(a) 2 (b) 10 (c) 4 (d) 5
Q.71. If , then ?????? ? - ?????? ? = 2 ?????? ?
is: ( ?????? ? + ?????? ?)
SSC CPO 03/10/2023 (1st Shift)
(a) (b) - 2 ?????? ? 2 ?????? ?
(c) (d) 2 ?????? ? - 2 ?????? ?
Q.72. The value of
is: ( 1 + ??????
4
?? - ??????
4
?? ) ??????????
2
??
SSC CPO 03/10/2023 (2nd Shift)
(a) -1 (b) 1 (c) -2 (d) 2
Q.73. If , and ?????? ? =
4
3
, 0 < ? <
p
2
, then the value of 5 ????????
2
? ?????? ? = ??????
2
?
p is :
SSC CPO 03/10/2023 (2nd Shift)
(a) (b) (c) (d)
7
27
5
27
25
27
125
27
Q.74. If , then ?????? ?? =
15
17
, 0 = ?? = 90°
the value of is : ?????? ( 90° - ?? )
SSC CPO 03/10/2023 (2nd Shift)
(a) (b) (c) (d)
8
15
7
15
2
15
2 2
15
Q.75. If sin 2A = cos 15°, then the
smallest positive value of A is:
SSC CPO 03/10/2023 (3rd Shift)
(a) 7.5° (b) 47.5° (c) 75° (d) 37.5°
Q.76. If A + B = 90°, then the expression
+ cos² A + cos² B is equal to:
?????? ??
?????? ??
SSC CPO 03/10/2023 (3rd Shift)
(a) cot² B (b) cosec² A
(c) cosec² B (d) cos² A
Q.77. In ?ABC, ?B = 90° and AB : BC= 1 :
2. The value of cos A + tan C is:
SSC CPO 04/10/2023 (1st Shift)
(a) (b)
1 + 5
2 5
2 5
2 + 5
(c) (d)
5 + 5
2 5
2 + 5
2 5
Q.78. If sin A + , then the value ??????
2
?? = 1
of is : ??????
4
?? + ??????
6
??
SSC CPO 04/10/2023 (1st Shift)
(a) 1 (b) cos A (c) sin A (d) 0
Q.79. If , then ?nd ?????? ?? - ?????? ?? = ??
the value of sec A.
SSC CPO 04/10/2023 (2nd Shift)
(a) (b)
??
2
- 1
??
2
+ 1
??
2
+ 1
??
2
- 1
(c) (d)
??
2
+ 1
2 ??
??
2
+ 1
??
Q.80. 2(sin 1° sec 89°) + 3 (cos 11° × ×
cosec 79°) + 5 (tan 21° tan 69°) = ? ×
SSC CPO 04/10/2023 (2nd Shift)
(a) 20 (b) 12 (c) 11 (d) 10
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Page 5
Pinnacle Day: 11th - 18th Trigonometry
Trigonometry
It is that branch of Mathematics which
deals with the measurements of the
sides and angles of a triangle and also
the problems associated with angles.
Important Formulae
In a right angled triangle, as shown below
sin = • ?
??
??
• ?????????? ? =
??
??
• ?????? ? =
??
??
• ?????? ? =
??
??
• ?????? ? =
??
??
• ?????? ? =
??
??
Note that
= , ?????????? ?
1
?????? ?
? ?????????? ? × ?????? ? = 1
Similarly, = ?????? ?
1
?????? ?
? ?????? ? × ?????? ? = 1
And, = ?????? ?
1
?????? ?
? ?????? ? × ?????? ? = 1
Also, ?????? ? =
?????? ?
?????? ?
? sin = cos (90° - ) ? ?
? cosec = sec (90° - ) ? ?
? tan = cot (90° - ) ? ?
Conversion of trigonometric ratios
? sin (-x) = - sin x
? cosec (-x) = - cosec x
? cos (-x) = cos x
? sec (-x) = sec x
? tan (- x) = - tan x
? cot (- x) = - cot x
Signs of the trigonometric ratios in
different quadrants :
x axis no change • ± ? ?
y axis changes like • ± ? ?
sin ? ? ?????? ?
tan ? ? ?????? ?
cosec ? ? ?????? ?
Important Identities
? ??????
2
? + ??????
2
? = 1
? ??????
2
? - ??????
2
? = 1
? ??????????
2
? - ??????
2
? = 1
We can derive all other identities from
these three basic identities.
Values of Trigonometric functions for
different values of . ?
The value of sin or cos never exceeds ? ?
1, whereas the value of sec or cosec is ? ?
always greater than or equal to 1.
Range of all trigonometric functions
Minimum Maximum
Sinx,cosx
(odd powers)
-1 + 1
Sinx,cosx
(even powers)
0 +1
Tanx,cotx
(odd powers)
- 8 + 8
Tanx,cotx
(even powers)
0 + 8
Secx,cosecx
(odd powers)
- 8 + 8
Secx,cosecx
(even powers)
+1 + 8
Maximum and Minimum value of the
trigonometric expressions of the
different form :
a sinx • ± ?? ????????
Maximum value = ??
2
+ ??
2
Minimum value = - ??
2
+ ??
2
a + b • ???? ??
2
?? ???? ??
2
??
If, a b Maximum value = a >
Minimum value = b
If a < b Maximum value = b
Minimum value = a
a + b • ???????? ??
2
?? ???? ??
2
??
Maximum value = 8
Minimum value = ( + ?? ?? )
2
a + b • ???? ??
2
?? ???????? ??
2
??
a + b • ???? ??
2
?? ???? ??
2
??
a + b • ???? ??
2
?? ???? ??
2
??
Maximum and Minimum value of these
three trigonometric expression is 8 ??????
2 respectively. ????
+ • ???? ??
2 ??
?? ???? ??
2 ??
??
Maximum value = 1
Minimum value = (use x = 45 ° )
Example : Find the minimum and
maximum value of + ???? ??
2
?? ???? ??
4
?? ?
Maximum value of + = 1 ???? ??
2
?? ???? ??
4
??
Minimum value = + = ???? ??
2
45° ???? ??
4
45°
+ =
1
2
1
4
3
4
sinx cosx • ( )
??
Maximum value = 8
Minimum value = (
1
2
)
??
Example : Find the minimum value of
? ???? ??
4
?? ???? ??
4
??
Minimum value = = (
1
2
)
4 1
16
Compound Formulae
? ?????? ?? + ?? ( )
= ?????? ?? . ?????? ?? + ?????? ?? . ?????? ??
? ?????? ?? - ?? ( )
= ?????? ?? . ?????? ?? - ?????? ?? . ?????? ??
? ?????? ?? + ?? ( )
= ???????? . ???????? - ???????? . ????????
? ?????? ?? - ?? ( )
= ???????? . ???????? + ???????? . ?????? ??
? sin 2A = 2 sin A cos A
=
2 ?????? ??
1 + ??????
2
??
? cos 2A = ??????
2
?? - ??????
2
??
= 1 - 2 ??????
2
??
? cos2A =
1 - ??????
2
??
1 + ??????
2
??
= 2 ??????
2
?? - 1
Tan 2A = •
2 ????????
1 - ??????
2
??
? sin 3A = 3 sinA – 4 ??????
3
??
? cos 3A = 4 ??????
3
?? - 3 ????????
? ?????? 3 ?? =
3 ?????? ?? – ??????
3
??
1 - 3 ??????
2
??
? ?????? ? . ?????? 2? . ?????? 4?
=
1
4
?????? 3?
? ?????? ? . ?????? 2? . ?????? 4?
=
1
4
?????? 3?
? ?????? ?. ?????? 2? . ?????? 4? = ?????? 3?
? Tan(A + B) =
???????? + ????????
1 - ( ???????? × ???????? )
? Tan(A - B) =
???????? - ????????
1 + ( ???????? × ???????? )
? Cot (A + B )=
???????? ???????? - 1
???????? + ????????
? Cot (A - B) =
???????? ???????? + 1
???????? - ????????
Transformations of sums or
differences into products
sin C + sin D = 2 sin cos •
?? + ??
2
?? - ??
2
sin C - sin D = 2 cos sin •
?? + ??
2
?? - ??
2
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Pinnacle Day: 11th - 18th Trigonometry
cos C + cos D = 2 cos •
?? + ??
2
??????
?? - ??
2
cos C - cos D = - 2 sin sin •
?? + ??
2
?? - ??
2
Concepts
? a sinX + b cosX = m
a cosX - b sinX = n
Then, + = + ??
2
??
2
??
2
??
2
Example : if 3 sin Find ? + 5 ?????? ? = 5 .
the value of 3cos - 5sin ? ? ?
3cos - 5sin ? ? = 5
2
+ 3
2
- 5
2
=± 3
? a secX + b tanX = m
a tanX - b secX = n
Then, - = - ??
2
??
2
??
2
??
2
If secx and tanx are replaced by cosecx
and cotx respectively . Then the outcome
will be the same.
Example : if 29 sec ? - 21 ?????? ? = 20
Find the value of 29 tan ? ? - 21 ?????? ?
Sol:-- = 29 ?????? ? - 21 ?????? ?
= 0 = 2 0
2
- 2 9
2
+ 2 1
2
? sec x + tan x =
1
?????? ?? - ?????? ??
? cosec x + cot x =
1
?????????? ?? - ?????? ??
Some Important results
(i). If A + B + C = 180 . then, °
? tanA + tanB + tanC = tanA tanB tan C
? cotA cotB + cotB cotC + cotC cotA = 1
(ii). If A + B + C = 90 . then, °
? tanA tanB + tanB tanC + tanC tanA = 1
? cotA + cotB + cotC = cotA cotB cotC
(iii). If A + B = 45 . then °
? (1 + tanA) (1 + tanB) = 2
? (1 - cotA) (1 - cotB) = 2
Variety Questions
Q.1. If , then ?nd ??????
4
?? + ?????? ² ?? = 1
the value of ??????
4
?? - 3 ?????? ² ?? .
SSC CHSL Tier II 02/11/2023
(a) 1 (b) 2 (c) 2 (d) 1 - -
Q.2 If cot
2
? = 1 - e
2
, then the value of
cosec ? + cot
3
? sec ? is:
SSC CPO 05/10/2023 (3rd Shift)
(a) (b) ( 2 - ??
2
)
3
2
( 1 - ??
2
)
(c) (d) ( 1 - ??
2
)
3
2
( 2 - ??
2
)
1
2
Q.3. If cosecA + cotA = a , then ?nd ??
the value of .
( ??
2
?? - 1 )
( ??
2
?? + 1 )
SSC CHSL 17/08/2023 (2nd Shift)
(a) cosA (b) tanA (c) (d)
1
????????
1
????????
Q.4. If and ?????? ? + ?????? ? = ?? ?????? ? -
, then which of the relation ?????? ? = ??
satis?es for given values of p and q ?
SSC CHSL 11/08/2023 (4th Shift)
(a) (b) ??
2
- ??
2
= 4 ???? ??
2
- ??
2
= 8 ????
(c) (d) ??
2
+ ??
2
= 4 ???? ??
2
+ ??
2
= 4 ????
Q.5. If is an acute angle and sin cos ? ? ?
= 2 cos
3
- cos , then the value of sin ?
1
4
?
is: ?
SSC CHSL 10/08/2023 (2nd Shift)
(a) (b)
15 - 1
8
15 - 1
4
(c) (d)
15 + 1
4
15 - 1
2
Q.6. Simplify the following: 7(cot 11° cot
13° cot 19° cot 29° cot 61°cot 71° cot 77°
cot 79°)
SSC CHSL 09/08/2023 (3rd Shift)
(a) 7 (b) - 7 (c) - 1 (d) 1
Q.7. The given expression is equal to :
??????
4
?? + ??????
4
??
1 - 2 ??????
2
?? ??????
2
??
SSC CHSL 09/08/2023 (2nd Shift)
(a) 1 (b) -1 (c) 0 (d) 2
Q.8. If cot A + cos A = p and cot A - cos A
= q, then which of the following relation
is correct?
SSC CHSL 03/08/2023 (4th Shift)
(a) = p
2
+ q
2
(b) = p² + q
2
1
16 ????
1
4 ????
(c) = p² - q² (d) 4pq = p² + q² 16 ????
Q.9. The value of the expression 4( ??????
6
A + A) - 6 ( A + A) + 8 is : ??????
6
??????
4
??????
4
SSC CHSL 02/08/2023 (4th Shift)
(a) 4 (b) 8 (c) 7 (d) 6
Q.10. Evaluate the following:
2 + 2 + 2 + 2 ?????? 8 ?
SSC CGL 27/07/2023 (3rd shift)
(a) 2cos (b) 2cos2 (c) sin2 (d) cos2 ? ? ? ?
Q.11. Find the value of
1 - ?????? 3 ?
1 + ?????? 3 ?
SSC CGL 21/07/2023 (2nd shift)
(a) sec 3 - tan 3 (b) (sec 3 - tan 3 )
3
? ? ? ?
(c) (sec 3 - tan 3 )
2
(d) sec 3 + tan 3 ? ? ? ?
Q.12. The solution of the equation cos
2
?
+ sin + 1 = 0 lies between : ?
Graduate Level 30/06/2023 (shift 1)
(a) ( , ) (b) ( , )
3 p
4
5 p
4
5 p
4
7 p
4
(c) ( , ) (d) (- , )
p
4
3 p
4
p
4
p
4
Q.13. What is the value of (1+ cotA +
tanA) (sinA - cosA) ( ) ?
???????? ????????
?????? ³ ?? - ?????? ³ ??
SSC CHSL Tier II (26/06/2023)
(a) 1 (b) -1 (c) 0 (d) 2
Q.14. If sin m + sin n = p, cos m + cos n =
q, then ?nd the value of sin m sin n + ×
cos m cos n. ×
SSC CHSL 20/03/2023 (2nd Shift)
(a) p
2
+ q
2
- 2 (b)
( ??
2
+ ??
2
- 2 )
2
(c) p + q - pq (d) p + q + pq
Q.15. If a cot + b cosec = p and b cot ? ?
+ a cosec = q then p² - q
2
is equal to ? ?
_____.
SSC CGL Tier II (07/03/2023)
(a) – (b) – (c) b – a (d) + ??
2
??
2
??
2
??
2
??
2
??
2
Q.16. If A is an acute angle, the
simpli?ed form of
?????? ( p - ?? ). ?????? (
??
2
+ ?? ) ?????? (- ?? )
?????? ( p + ? ) ?????? (
3??
2
+ ? ) ?????? ( 2p - ?? )
SSC CGL 13/12/2022 (4th Shift)
(a) Cos²A (b) Sin A (c) Sin²A (d) Cos A
Q.17. What is the value of the expression
cos 2A cos 2B + sin
2
(A - B) - sin²(A + B) ?
SSC CGL 12/12/2022 (1st Shift)
(a) sin (2A - 2B) (b) sin (2A + 2B)
(c) cos (2A + 2B) (d) cos (2A - 2B)
Q.18. If = sin
2
60° +
?? - ?? ??????
2
30°
1 + ??????
2
30°
4tan
2
45° - cosec² 60°, then the value of k
(correct to two decimal places) is:
SSC CGL 03/12/2022 (3rd Shift)
(a) 5.55 (b) - 6.83 (c) - 5.58 (d) 6.83
Q.19. If Cos ? + Sin ? = Cos ?, ?nd the 2
value of (cos ? - Sin?).
SSC CPO 10/11/2022 (Afternoon)
(a) Sin ? (b) Cos ? 2 2
(c) Sin ? (d) Cos ?
1
2
1
2
Q.20. The value of is: ?????? 73° + ?????? 137°
SSC CHSL 10/06/2022 (Evening)
(a) sin 13 (b) cos 13 ° °
(c) cos 18 (d) sin 18 ° °
Q.21. The value of sin18° is given as
. Using this value, ?nd the value
5 - 1
4
of cos18°.
SSC CHSL 30/05/2022 (Afternoon)
(a) (b)
10 + 2 5
4
5 + 1
4
(c) (d)
6 + 2
4
10 - 2 5
4
Q.22. The value of is:
?????? 4 ? + ?????? 2 ?
?????? 4 ? + ?????? 2 ?
SSC CHSL 27/05/2022 ( Morning )
(a) (b) (c) (d) ?????? 3 ? ?????? 3 ? ?????? ? ?????? ?
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Pinnacle Day: 11th - 18th Trigonometry
Q.23. The value of - + ??????
2
46° ??????
2
44°
( ??????
2
1° + ??????
2
3° + ??????
2
5° +... + ??????
2
89° )
SSC CHSL 10/08/2021 (Evening)
(a) 23.5 (b) 20.5 (c) 22.5 (d) 21.5
Q.24. If A lies between 45° and 540°, and
sinA = 0.5, what is the value of in
??
3
degree ?
SSC CHSL 15/04/2021 (Morning)
(a) 170° (b) 175° (c) 165° (d) 160°
Q.25. If cos 27° = x, then the value of tan
63° is:
SSC CHSL 26/10/2020 (Evening)
(a) (b)
1 + ??
2
??
??
1 + ??
2
(c) (d)
1 - ??
2
??
??
1 - ??
2
Q.26. If ?? ??????
3
? + ?? ??????
3
? = ?????? ? ?????? ?
then the value of ?????? ?? ?????? ? = ?? ?????? ?,
is: ??
2
+ ??
2
SSC CHSL 20/10/2020 (Morning)
(a) 0 (b) 4 (c) 1 (d) 2
Q.27. If cos
2
? + cos
4
? = 1, then the value
of sin? + sin
2
? is:
SSC CHSL 16/10/2020 (Morning)
(a) 0 (b) (c) 2 (d) 1
1
2
Q.28. If sin? - cos? = , then ?nd the
7
17
value of sin? + cos?.
SSC CHSL 14/10/2020 (Evening)
(a) (b) (c) (d)
8
17
23
13
23
17
8
13
Q.29. If sec and ? ?????? ? ( 0° < ? < 90° )
are the roots of the equation
, then the value of k 6 ??
2
- ???? + 6 = 0
is:
SSC CHSL 14/10/2020 (Morning)
(a) (b) 3 (c) 2 (d) 3 3 2 3 3
Q.30. If cosA, sinA, cotA are in geometric
progression, then the value of tan
6
A -
tan
2
A is:
SSC CHSL 13/10/2020 (Afternoon)
(a) (b) 3 (c) (d) 1
1
2
1
3
Q.31. Find the value of Cos 225°
SSC CHSL 17/03/2020 (Afternoon)
(a) -0.7071 (b) -0.866
(c) 0.7071 (d) 0.866
Q.32. If 0° < A, B < 45°, cos(A + B) =
24
25
and sin(A - B) = , then tan2A = ?
15
17
SSC CGL 06/03/2020 (Evening)
(a) 0 (b) 1 (c) (d)
416
87
213
4
Q.33. If = then ??????
2
? - ??????
2
? ??????
2
?,
which of the following is true?
SSC CPO 13/12/2019 (Evening)
(a) cos cos = 1 ? ?
(b) cos cos = ? ? 2
(c) - = ??????
2
? ??????
2
? ??????
2
?
(d) - = ??????
2
? ??????
2
? ??????
2
?
Q.34 . everything between 0° and 90° For
it is given that a '
??
?? , ( ?? = 1 , 2 , 3 ......... 20 )
sin + sin + sin …………...sin = 20. a
1
a
2
a
3
a
20
Find the value (in degrees) of ( a
1
+
) . a
2
+ a
3
.... a
20
SSC CHSL 11/07/2019 (Morning)
(a) 1800° (b) 900° (c) 0° (d) 20°
Q.35. For and both being acute a ß
angles, it is given that sin( + ) = 1, cos a ß
( - ) = . The values of and are : a ß
1
2
a ß
SSC CHSL 10/07/2019 (Evening)
(a) (b) 75° , 15° 45° , 15°
(c) (d) 75° , 45° 60° , 30°
Q.36. If = , then what is the value of ? 9°
cot .cot .cot .cot .cot .cot .cot ? 2 ? 3 ? 4 ? 5 ? 6 ?
.cot .cot ? 7 ? 8 ? 9 ?
SSC CHSL 10/07/2019 (Evening)
(a) - 1 (b) 1 (c) (d) 3 3
1
3
Q.37. Which among the following is an
irrational quantity ?
SSC CHSL 10 /07/ 2019 (Afternoon)
(a) tan tan (b) sin 30° . 60° 30°
(c) tan (d) cos 45° 30°
Q.38. Which among the following
increases continuously in the range
? 0° ? ? ? 90°
SSC CHSL 09/07/2019 (Evening)
(a) cot (b) cosec (c) tan (d) cos ? ? ? ?
Q.39. If sec = 4 and tan = , ( ) ? ?? ?
4
??
?? ? 0
then the value of 8( ) is : ??
2
-
1
??
2
SSC CHSL 05/07/2019 (Afternoon)
(a) (b) (c) (d)
1
16
1
4
1
2
1
8
Q.40. If ×
1 - ?????? ?
1 + ?????? ?
?????????? ? - ?????? ?
?????????? ? + ?????? ?
= , then the value of r is:
1 - ??
1 + ??
SSC CHSL 03/07/2019 (Afternoon)
(a) Sin ?? (b) Cosec ?? (c) Sec ?? (d) Cos ??
Q.41. If =
( ?????? ? - ?????????? ?)( ?????? ? - ?????? ?)
??????
2
? - ??????
2
?
??
3
, then r = ?
SSC CHSL 03/07/2019 (Morning)
(a) Sin Cos (b) tan ? ? ?
(c) Cot (d) Cosec Sec ? ? ?
Q.42. If 3 - 2 - 3Cos ?? = 0, 0° < ?? < ??????
2
?
90° , then what is the value of (2Cosec ?? +
tan ?? ) ?
SSC CHSL 02/07/2019 (Evening)
(a) 7v3 (b) 5v3 (c) (d)
5 3
3
7 3
3
Q.43. A simpli?ed of
( + )( ) is :
?????? ?
1 + ?????? ?
1 + ?????? ?
?????? ?
1
?????? ? + ?????? ?
SSC CHSL 02/07/2019 (Evening)
(a) Cos (b) 2Sin (c) Sin (d) 2Cos ? ? ? ?
Q.44. The value of
is:
( ?????? 9° + ?????? 81° )( ?????? 9° + ?????????? 81° )
?????? 56° . ?????? 34° + ?????? 25° . ?????????? 65°
SSC CHSL 02/07/2019 (Evening)
(a) (b) 4 (c) 2 (d) 1
1
2
Q.45. is
( ?????? ? + ?????? ?) ( 1 - ?????? ?)
?????????? ?( 1 + ?????? ?) ( ?????????? ? - ?????? ?)
equal to:
SSC CHSL 02/07/2019 (Morning)
(a) Sin (b) Sec (c) Cos (d) Cosec ? ? ? ?
Q.46. If 3cos
2
A + 7sin
2
A = 4, then what is
the value of cot A, given that A is an
acute angle ?
SSC CGL 13/06/2019 (Evening)
(a) 1 (b) (c) (d) 3
3
2
1
3
Q.47. If Cosec = , then Sin + Cos ?
13
12
? ?
- tan is equal to : ?
SSC CGL 11/06/2019 (Afternoon)
(a) (b) (c) (d) -
91
65
139
65
71
65
71
65
Q.48. The value of ??????
2
38° + ??????
2
52° +
is equal to: ??????
2
30° - ??????
2
45°
SSC CGL 10/06/2019 (Afternoon)
(a) (b) (c) (d)
1
3
1
4
3
4
1
2
Q.49. If Sec4 = Cosec ( + 20°) , then ? ? ?
is equal to:
SSC CGL 10/06/2019 (Afternoon)
(a) 22° (b) 18° (c) 14° (d) 20°
Q.50. If 3 sin = 2 0° < ?? < 90°, then ? ??????
2
?,
the value of ( ??????
2
? + ??????
2
? - ???? ??????
2
?)
is :
SSC CGL 10/06/2019 (Morning)
(a) -2 (b) - (c) (d) 2
7
3
7
3
Q.51. If , then k = ?
?????? ? + ?????? ?
?????? ? - ?????? ?
=
?? + 1
?? - 1
SSC CGL 10/06/2019 (Morning)
(a) cosec (b) sec (c) cos (d) sin ? ? ? ?
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Pinnacle Day: 11th - 18th Trigonometry
Q.52. If = 2
1
?????????? ? - 1
+
1
?????????? ? + 1
sec?? , 0° < ?? < 90° , then the value of
(cot?? + cos ?? ) is:
SSC CGL 07/06/2019 (Evening)
(a) (b)
1 + 2
2
2 + 2
2
(c) (d)
2 + 3
2
1 + 2
Q.53. If sin = , 0° < < 90°, ?
??
??
2
+ ??
2
?
then the value of sec + tan is : ? ?
SSC CGL 07/06/2019 (Afternoon)
(a) (b)
??
2
+ ??
2
+ ??
??
??
2
+ ??
2
+ ??
2 ??
(c) (d)
??
2
+ ??
2
+ ??
2 ??
??
2
+ ??
2
+ ??
??
Q.54. If - 3cos + 2 = , 0° < < ??????
2
? ? ??????
2
? ?
90°, then the value of 2cosec + 4cot is : ? ?
SSC CGL 07/06/2019 (Afternoon)
(a) (b) (c) (d)
8 3
3
4 3
4
2 3 4 3
Q.55. The value of ,
1
???????? - ????????
-
1
????????
0° < x < 90°, is equal to :
SSC CGL 07/06/2019 (Morning)
(a) 2secx (b) tanx (c) 2cosx (d) cotx
Q.56. If - 3sec + 3 = 0 , 0° < x < ??????
2
? ?
90°, then the value of sin + cot is : ? ?
SSC CGL 07/06/2019 (Morning)
(a) (b) (c) (d)
5 3
6
2 3
5 3
3
3 3
Q.57. If (1 + ??????
2
?) + ( 1 + ( ??????
2
?)
- 1
)
= k, then = ? ??
SSC CGL 06/06/2019 (Evening)
(a) cosec .sec (b) cosec .cos ? ? ? ?
(c) sin .cos (d) sin .sec ? ? ? ?
Q.58. The value of is
1
?????? ?
-
??????
2
?
1 + ?????????? ?
SSC CGL 06/06/2019 (Evening)
(a) 0 (b) 1 (c) 2 (d) -1
Q.59. ( ) = ?
1
1 + ??????
2
?
+
1
1 + ??????????
2
?
SSC CGL 06/06/2019 (Evening)
(a) (b) 1 (c) (d) 2 ??????
2
? ??????????
2
?
Q.60. If = 4, 0° < ??
?????? ?
1 - ?????? ?
+
?????? ?
1 + ?????? ?
< 90° , then the value of (tan ?? + cosec?? )
is :
SSC CGL 06/06/2019 (Evening)
(a) (b) (c) (d)
5 2
2
5 3
3
4 3
3
5 2
3
Q.61. = ?
?????? ? - ?????? ? + 1
?????? ? + ?????? ? - 1
SSC CGL 06/06/2019 (Morning)
(a) sec sin (b) sec tan ? ? ? ?
(c) sec + tan (d) sec - tan ? ? ? ?
Q.62. The value of
is ??????
2
? + ??????????
2
? × ??????
2
? - ??????
2
?
equal to :
SSC CGL 06/06/2019 (Morning)
(a) (b) ?????????? ? ??????
2
? ?????? ? ??????
2
?
(c) sin (d) ? ??????
2
? ?????????? ? ??????
2
?
Q.63. If sec - tan = P , then cosec = ? ? ? ?
SSC CGL 04/06/ 2019 (Evening)
(a) (b)
2 ??
1 - ??
2
1 - ??
2
1 + ??
2
(c) (d)
??
2
+ 1
1 - ??
2
2 ??
1 + ??
2
Q.64. If and 0
0
< ? < 90
0
??????
2
?
then the value of ( = 3 ( ??????
2
? - ??????
2
?)
1
2
)
- 1
is : ?????? ? + ?????? ?
SSC CGL 04/06/2019 (Afternoon)
(a) (b) 2(2 - 3 + 2 3 )
(c) 2( (d) 3 - 1 ) 3 + 1
Q.65. is (
?????? ? - 2 ??????
3
?
2 ??????
3
? - ?????? ?
)
2
+ 1 , ? ? 45
0
equal to.
SSC CGL 04/06/2019 (Afternoon)
(a) (b) (c) (d) 2 ??????????
2
? ??????
2
? ??????
2
? ??????
2
?
Q.66. (cosecA - sinA + (secA - cosA - )
2
)
2
(cotA - tanA is equal to: )
2
SSC CPO 16/03/2019 (Evening)
(a) 2 (b) 0 (c) 1 (d) -1
Q.67. Find the value of 1 +
??????
2
??
1 + ?????? ??
SSC CPO 15/03/2019 (Evening)
(a) cosec A (b) cos A (c) sec A (d) sin A
Q.68. If x = a cos + b sin and y = a sin - ? ?
b cos , the value of + is: ? ??
2
??
2
SSC CPO 15/03/2019 (Morning)
(a) - (b) a - b (c) + (d) a + b ??
2
??
2
??
2
??
2
Practice Questions
SSC CPO 2023 Tier - 1
Q.69. Find the value of (tan
2
? + tan
4
?).
SSC CPO 03/10/2023 (1st Shift)
(a) (b) ??????
2
? - ??????
4
? ??????
2
? - ??????
2
?
(c) (d) ??????
4
? + ??????
2
? ??????
4
? - ??????
2
?
Q.70. If ? is an acute angle and sin ? +
cosec ? = 2,then the value of ? + ??????
5
? is: ??????????
5
SSC CPO 03/10/2023 (1st Shift)
(a) 2 (b) 10 (c) 4 (d) 5
Q.71. If , then ?????? ? - ?????? ? = 2 ?????? ?
is: ( ?????? ? + ?????? ?)
SSC CPO 03/10/2023 (1st Shift)
(a) (b) - 2 ?????? ? 2 ?????? ?
(c) (d) 2 ?????? ? - 2 ?????? ?
Q.72. The value of
is: ( 1 + ??????
4
?? - ??????
4
?? ) ??????????
2
??
SSC CPO 03/10/2023 (2nd Shift)
(a) -1 (b) 1 (c) -2 (d) 2
Q.73. If , and ?????? ? =
4
3
, 0 < ? <
p
2
, then the value of 5 ????????
2
? ?????? ? = ??????
2
?
p is :
SSC CPO 03/10/2023 (2nd Shift)
(a) (b) (c) (d)
7
27
5
27
25
27
125
27
Q.74. If , then ?????? ?? =
15
17
, 0 = ?? = 90°
the value of is : ?????? ( 90° - ?? )
SSC CPO 03/10/2023 (2nd Shift)
(a) (b) (c) (d)
8
15
7
15
2
15
2 2
15
Q.75. If sin 2A = cos 15°, then the
smallest positive value of A is:
SSC CPO 03/10/2023 (3rd Shift)
(a) 7.5° (b) 47.5° (c) 75° (d) 37.5°
Q.76. If A + B = 90°, then the expression
+ cos² A + cos² B is equal to:
?????? ??
?????? ??
SSC CPO 03/10/2023 (3rd Shift)
(a) cot² B (b) cosec² A
(c) cosec² B (d) cos² A
Q.77. In ?ABC, ?B = 90° and AB : BC= 1 :
2. The value of cos A + tan C is:
SSC CPO 04/10/2023 (1st Shift)
(a) (b)
1 + 5
2 5
2 5
2 + 5
(c) (d)
5 + 5
2 5
2 + 5
2 5
Q.78. If sin A + , then the value ??????
2
?? = 1
of is : ??????
4
?? + ??????
6
??
SSC CPO 04/10/2023 (1st Shift)
(a) 1 (b) cos A (c) sin A (d) 0
Q.79. If , then ?nd ?????? ?? - ?????? ?? = ??
the value of sec A.
SSC CPO 04/10/2023 (2nd Shift)
(a) (b)
??
2
- 1
??
2
+ 1
??
2
+ 1
??
2
- 1
(c) (d)
??
2
+ 1
2 ??
??
2
+ 1
??
Q.80. 2(sin 1° sec 89°) + 3 (cos 11° × ×
cosec 79°) + 5 (tan 21° tan 69°) = ? ×
SSC CPO 04/10/2023 (2nd Shift)
(a) 20 (b) 12 (c) 11 (d) 10
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Pinnacle Day: 11th - 18th Trigonometry
Q.81. Find the value of .
1 - ????????
1 + ?????? ??
SSC CPO 04/10/2023 (2nd Shift)
(a) (b)
1 + ?????? ??
?????? ??
1 + ?????? 2 ??
?????? 2 ??
(c) (d)
1 - ?????? ??
?????? ??
1 - ?????? 2 ??
?????? 2 ??
Q.82. then ?nd 7 ??????
2
?? + 3 ??????
2
?? = 4 ,
cotA :
SSC CPO 04/10/2023 (2nd Shift)
(a) (b) (c) (d)
1
3
1
2
3 2
Q.83. Find the least value of 16cosec
2
? +
25sin
2
?.
SSC CPO 04/10/2023 (3rd Shift)
(a) 35 (b) 38 (c) 42 (d) 40
Q.84. If cosec A + cot A = 3, 0 = A = 90°,
then ?nd the value of cosA.
SSC CPO 04/10/2023 (3rd Shift)
(a) (b) (c) (d)
2
5
3
4
3
5
4
5
Q.85. The value of is :
1 + ?????? ??
1 - ?????? ??
SSC CPO 04/10/2023 (3rd Shift)
(a) cosec A + cotA (b) cosec A - cotA
(c) sec A - tanA (d) sec A + tanA
Q.86. If sin A = , then ?nd the value of (
2
3
)( ). 7 - ?????? ?? 3 + ?????? ??
SSC CPO 05/10/2023 (1st Shift)
(a) (b)
61
3
+
17
5
61
3
-
17
3 5
(c) (d)
61
3
+
17
3 5
61
3 5
+
17
3
Q.87. If sin A = , 0 < A < 90°, then ?nd
3
2
the value of 2(cosec A + cot A).
SSC CPO 05/10/2023 (1st Shift)
(a) (b) (c) (d)
1
3
3
2
3
2 3
Q.88. If sin = , is ß
1
3
( ?????? ß - ?????? ß )
2
equal to :
SSC CPO 05/10/2023 (1st Shift)
(a) (b) (c) (d)
1
3
1
2
2
3
3
4
Q.89. Find the value of
+ 3tan25
??????
2
39° + ??????
2
( 90° - 39° )
??????
2
35° + ??????
2
( 90° - 35° )
°
tan65 . °
SSC CPO 05/10/2023 (2nd Shift)
(a) 1 (b) 3 (c) 4 (d) 2
Q.90. If tan? + cot? = 2, ? is an acute
angle, then ?nd the value of 2 tan
25
? +3
cot
20
? + 5 tan
30
? cot
15
?
SSC CPO 05/10/2023 (2nd Shift)
(a) 12 (b) 10 (c) 8 (d) 6
Q.91. Find the value of sin (50° + A) - cos
(40° - A).
SSC CPO 05/10/2023 (2nd Shift)
(a) 1 (b) 0 (c) -1 (d) 2
Q.92. If sin ? + cos ? = , then what is
11
3
sin? - cos? ?
SSC CPO 05/10/2023 (2nd Shift)
(a) (b) (c) (d)
5
3
7
3
7
4
5
2
Q.93. If , then ?????? ? + ?????? ? = 2 ?????? ?
?nd :
?????? ? - ?????? ?
?????? ?
SSC CPO 05/10/2023 (3rd Shift)
(a) (b) 1 (c) 1 (d) - 2 - 2
SSC CHSL 2023 Tier - 1
Q.94. If sinA + A = 1, then the value ??????
2
of the expression ( A + A) is ??????
2
??????
4
___________
SSC CHSL 02/08/2023 (1st Shift)
(a) (b) 1 (c) 2 (d)
3
2
1
2
Q.95. If cot = , ?nd the value of ?
9
17
. ??????????
2
?
SSC CHSL 02/08/2023 (1st Shift)
(a) 1 (b) (c) (d)
9
17
208
289
8
17
1
81
289
Q.96. is equal to :
?????? ?
?????? ? - 1
+
?????? ?
?????? ? + 1
SSC CHSL 02/08/2023 (1st Shift)
(a) 2 (b) 2 ??????
2
? ??????
2
?
(c) 2 (d) ??????
2
? 2 ??????
2
?
Q.97. If A = 60°, B = 30°, then ?nd the
value of sin A cos B + cos A sin B.
SSC CHSL 02/08/2023 (2nd Shift)
(a) 0 (b) 12 (c) 13 (d) 1
Q.98. If (sin ) = 0, then the value ? - ?????? ?
of ( - ) is: ?????? (p - ?) + ??????
p
2
?
SSC CHSL 02/08/2023 (2nd Shift)
(a) 1 (b) 0 (c) (d) 3 2
Q.99. The value of cot 13° cot 27°cot 60°
cot 63° cot 77° is:
SSC CHSL 02/08/2023 (2nd Shift)
(a) (b) 0 (c) (d) 1
1
3
3
Q.100. If , then the ?????? ? + ?????? ? = 3
value of is : ??????
3
? + ??????
3
?
SSC CHSL 02/08/2023 (3rd Shift)
(a) 0 (b) (c) (d) 2
1
3
3 3
Q.101. If , where ?????? ( 3 a) = ?????? (a - 22° )
, then what is the value of ? 3 a < 90° a
SSC CHSL 02/08/2023 (3rd Shift)
(a) 29° (b) 26° (c) 27° (d) 28°
Q.102. If , then 2 ?????? ? - ?????? ? = ?? ?????? ?
= ?
SSC CHSL 02/08/2023 (3rd Shift)
(a) (b)
1 - ??
??
1 - ??
2
??
(c) (d)
1 + ??
2
??
??
2
- 1
??
Q.103. If cos A = , then ?nd the value
63
65
of tan A + cot A (up to two places of
decimal).
SSC CHSL 02/08/2023 (4th Shift)
(a) 4.19 (b) 2.76 (c) 3.19 (d) 5.23
Q.104. If +
?????? ?
( 1 + ?????? ?)
?????? ?
( 1 - ?????? ?)
= 4
and is acute , then the value of is : ? ?
SSC CHSL 03/08/2023 (1st Shift)
(a) 60° (b) 15° (c) 45° (d) 30°
Q.105. If then what ?????? a + ?????? a =
2
3
is equal to? ( ?????? a + ?????? a)
SSC CHSL 03/08/2023 (1st Shift)
(a) 1 (b) 2 (c) 6 (d) 0
Q.106. Find the value of
?????? 0° + ?????? 30° - ?????? 45° + ?????????? 60° +
. ?????? 90°
SSC CHSL 03/08/2023 (1st Shift)
(a) (b) (c) (d)
7
6 3
3
6
7
6
7
2 3
Q.107. The value of
is : ( ??????????
2
?? - 1 )( ??????
2
?? - 1 )
SSC CHSL 03/08/2023 (2nd Shift)
(a) 4 (b) 3 (c) 2 (d) 1
Q.108. If sin + cos = , ?nd the ? ?
1
29
value of .
?????? ? + ?????? ?
?????? ? - ?????? ?
SSC CHSL 03/08/2023 (2nd Shift)
(a) (b) (c) (d)
1
41
43
29
41
29
1
43
Q.109. = ________. ??????
2
35° + ??????
2
55°
SSC CHSL 03/08/2023 (3rd Shift)
(a) 0 (b) 1 (c) 2sin35° (d) 2sin55°
Q.110. The given expression is equal to :
( 1 + ??????
2
?? )
??????????
2
?? . ????????
SSC CHSL 03/08/2023 (3rd Shift)
(a) A (b) Sec A (c) Tan A (d) Tan
2
A ??????
2
Q.111. is equal to : ??????
4
?? - ??????
4
??
SSC CHSL 03/08/2023 (3rd Shift)
(a) 2 cos
2
A + 1 (b) 1 - 2 sin
2
A
(c) 2 sin
2
A - 1 (d) -(2 sin
2
A + 1)
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