NCERT Exemplar -Trigonometric Functions Notes | EduRev

JEE Revision Notes

JEE : NCERT Exemplar -Trigonometric Functions Notes | EduRev

The document NCERT Exemplar -Trigonometric Functions Notes | EduRev is a part of the JEE Course JEE Revision Notes.
All you need of JEE at this link: JEE

Q.1. Prove that 
NCERT Exemplar -Trigonometric Functions Notes | EduRev
Ans.
 NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
[Rationalizing the denominator]
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev R.H.S. Hence proved.

Q.2. If NCERT Exemplar -Trigonometric Functions Notes | EduRev then prove that 

NCERT Exemplar -Trigonometric Functions Notes | EduRev is also equal to y.
NCERT Exemplar -Trigonometric Functions Notes | EduRev
Ans.
Given that:
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev 
Hence proved.

Q.3. If m sin θ = n sin (θ + 2α), then prove that tan (θ + α) cot α = NCERT Exemplar -Trigonometric Functions Notes | EduRev 
[Hint: Express NCERT Exemplar -Trigonometric Functions Notes | EduRev and apply componendo and dividendo]
Ans.
Given that: m sin θ = n sin (θ + 2α)
    NCERT Exemplar -Trigonometric Functions Notes | EduRev
Using componendo and dividendo theorem we get
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev 
Hence proved.

Q.4. If cos (α + β) = NCERT Exemplar -Trigonometric Functions Notes | EduRev and sin (α – β) =NCERT Exemplar -Trigonometric Functions Notes | EduRev where α lie between 0 and NCERT Exemplar -Trigonometric Functions Notes | EduRev find the value of tan2α.
[Hint: Express tan 2 α as tan (α + β + α – β]
Ans.
Given that:
NCERT Exemplar -Trigonometric Functions Notes | EduRev      

NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
Now    tan 2α  = tan [α + β + α – β]
= tan [(α  + β) + (α  – β)]
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
Hence,  NCERT Exemplar -Trigonometric Functions Notes | EduRev

Q.5. If tan x = NCERT Exemplar -Trigonometric Functions Notes | EduRev  then find the value of  NCERT Exemplar -Trigonometric Functions Notes | EduRev
Ans.
Given that: tan x = NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
Hence, NCERT Exemplar -Trigonometric Functions Notes | EduRev

Q.6. Prove that cosθ cos NCERT Exemplar -Trigonometric Functions Notes | EduRev sin 7θ sin 8θ.
[Hint: Express L.H.S. =  NCERT Exemplar -Trigonometric Functions Notes | EduRev
Ans.
L.H.S. NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
= – sin 8θ sin (– 7θ) = sin 7θ sin 8θ     [∵ sin (-θ) = -sin θ]
L.H.S. = R.H.S. Hence proved.

Q.7. If  a cos θ + b sin θ = m and a sin θ – b cos θ = n, then show that a2 + b2 = m2 + n2
Ans.
Given that: a cos θ + b sin θ
= m and a sin θ  b cos θ = n
R.H.S. = m2 + n2 = (a cos θ + b sin θ)2 + (a sin θ – b cos θ)2
= a2 cos2 θ + b2 sin2 θ + 2ab sin θ cos θ + a2 sin2 θ + b2 cos2 θ  2ab sin θ cos θ
= a2 cos2 θ + b2  sin2 θ + a2  sin2 θ + b2  cos2 θ
= a2(cos2 θ + sin2 θ) + b2(sin2 θ + cos2 θ)
= a2.1 + b2.1 = a2 + b2  L.H.S.
L.H.S = R.H.S.
Hence    proved.

Q.8. Find the value of tan 22°30 ′ .
[Hint: Let θ = 45°, use NCERT Exemplar -Trigonometric Functions Notes | EduRev
Ans. 
Let 22°30’ NCERT Exemplar -Trigonometric Functions Notes | EduRev ∴ θ =45°
tan 22°30’ NCERT Exemplar -Trigonometric Functions Notes | EduRev
Put θ = 45°
NCERT Exemplar -Trigonometric Functions Notes | EduRev
Hence, tan 22°30’ = √2 - 1    

Q.9. Prove that sin 4A = 4sinA cos3A – 4 cosA sin3A.
Ans.
L.H.S. sin 4A = sin (A + 3A)
= sin A cos 3A + cos A sin 3A
= sin A(4 cosA – 3 cos A) + cos A(3 sin A – 4 sin3 A)
= 4 sin A cos3 A – 3 sin A cos A + 3 sin A cos A – 4 cos A sin3 A
= 4 sin A cos3 A – 4 cos A sin3 A. R.H.S.
L.H.S. = R.H.S.
Hence proved.

Q.10. If tanθ + sinθ = m and tanθ – sinθ = n, then prove that m2 – n2 = 4sinθ tanθ
[Hint: m + n = 2tanθ, m – n = 2 sinθ, then use m2 – n2 = (m + n) (m – n)]
Ans.
Given that: tan θ  + sin θ = m and tan θ – sin θ = n
L.H.S. m2 – n2 = (m + n)(m – n)
= [(tan θ + sin θ) + (tan θ – sin θ)]. [(tan θ + sin θ) – (tan θ – sin θ)]
= (tan θ + sin θ + tan θ – sin θ).(tan θ + sin θ – tan θ + sin θ)
= 2 tan θ.2 sin θ= 4 sin θ tan θ. R.H.S.
L.H.S. = R.H.S. Hence proved.

Q.11. If tan (A + B) = p, tan (A – B) = q, then show that tan 2 A = NCERT Exemplar -Trigonometric Functions Notes | EduRev
[Hint: Use 2A = (A + B) + (A – B)]
Ans.
Given that: tan (A + B) = p, tan (A – B) = q
tan 2A = tan (A + B + A – B) = tan [(A + B) + (A – B)]
NCERT Exemplar -Trigonometric Functions Notes | EduRev

Q.12. If cosα + cosβ = 0 = sinα + sinβ, then prove that cos 2α + cos 2β = – 2cos (α + β).
[Hint: (cosα + cosβ)2 – (sinα + sinβ)2 = 0]
Ans.
Given that: cos α + cos β = 0 ................(i)
and sin α + sin α = 0 ...............(ii)
From  (i) and (ii) we have
(cos α + cos β)2  (sinα + sin β)2 = 0
⇒(cos2 α + cos2 β  + 2 cos α cos  β) – (sin2 α + sin2  β + 2 sin α sin  β) = 0
⇒cos2 α + cos2  β + 2 cos α cos  β – sin2 α – sin2  β – 2 sin α sin  β = 0
⇒(cos2 α – sin2 α) + (cos2  β  – sin2  β) + 2(cos α cos  β – sin α sin  β) = 0
⇒cos2 α + cos2 β + 2 cos (α +  β) = 0
Hence, cos 2α + cos 2 β = – 2 cos (α +  β).
Hence proved.

Q.13. If  NCERT Exemplar -Trigonometric Functions Notes | EduRev then show that

NCERT Exemplar -Trigonometric Functions Notes | EduRev
 [Hint: Use Componendo and Dividendo].
Ans.
Given that:NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
(Using componendo and dividendo theorem)
NCERT Exemplar -Trigonometric Functions Notes | EduRev

Q.14. If tanθ = NCERT Exemplar -Trigonometric Functions Notes | EduRev  then show that sinα + cosα = √2 cosθ.
[Hint: Express tanθ = tan NCERT Exemplar -Trigonometric Functions Notes | EduRev
Ans.
Given that: NCERT Exemplar -Trigonometric Functions Notes | EduRev
⇒  NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
⇒   √2 cos θ = cos α + sin α
⇒ sin α + cos α = √2 cos θ.
Hence proved.

Q.15. If sinθ + cosθ = 1, then find the general value of θ.
Ans.
Given that: sin θ + cos θ  = 1
Dividing both sides by NCERT Exemplar -Trigonometric Functions Notes | EduRev we get
NCERT Exemplar -Trigonometric Functions Notes | EduRev  ...(1)
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
Hence, the general values of θ are 2nπ,n∈Z
Alternate method:
From eqn. (i) we get
NCERT Exemplar -Trigonometric Functions Notes | EduRev
Hence, the general value of NCERT Exemplar -Trigonometric Functions Notes | EduRev

Q.16. Find the most general value of θ satisfying the equation tanθ = –1 and
NCERT Exemplar -Trigonometric Functions Notes | EduRev
Ans.
Given that: tan θ = – 1 and cos θ = NCERT Exemplar -Trigonometric Functions Notes | EduRev
tan θ = – 1
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
Hence, the general solution is NCERT Exemplar -Trigonometric Functions Notes | EduRev

Q.17. If cotθ + tanθ = 2 cosecθ, then find the general value of θ.
Ans.
Given that: cot θ + tan θ = 2 cosec θ
NCERT Exemplar -Trigonometric Functions Notes | EduRev
 ⇒  2sin θ cos θ = sin θ
⇒ 2 sin θ cos θ  sin θ = 0
⇒ sin θ (2 cos θ  1) = 0
⇒ sin θ = 0 or 2 cos θ  1 = 0 or NCERT Exemplar -Trigonometric Functions Notes | EduRev
⇒  NCERT Exemplar -Trigonometric Functions Notes | EduRev
⇒  NCERT Exemplar -Trigonometric Functions Notes | EduRev
Hence, the general values of θ is NCERT Exemplar -Trigonometric Functions Notes | EduRev

Q.18. If 2sin2θ = 3cosθ, where 0 ≤ θ ≤ 2π, then find the value of θ.
Ans.
Given that: 2 sin2 θ = 3 cos θ
⇒ 2(1 – cos2 θ) = 3 cos θ
⇒ 2 – 2 cos2 θ – 3 cos θ = 0
⇒ 2 cos2 θ + 3 cos θ – 2 = 0
⇒ 2 cos2 θ+ 4 cos θ – cos θ – 2 = 0
⇒2 cos θ (cos θ + 2) – 1(cos θ + 2) = 0
⇒ (cos θ + 2) (2 cos θ – 1) = 0   
⇒ cos θ + 2 = 0 or 2 cos θ – 1 = 0  
⇒ cos θ  - 2 [- 1  cos θ  1]
 2 cos θ  1 = 0
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
∴  NCERT Exemplar -Trigonometric Functions Notes | EduRev
and  NCERT Exemplar -Trigonometric Functions Notes | EduRev
Hence, the value of θ are NCERT Exemplar -Trigonometric Functions Notes | EduRev

Q.19. If secx cos5x + 1 = 0, where 0 < x ≤NCERT Exemplar -Trigonometric Functions Notes | EduRev then find the value of x.
Ans.
Given that: sec x cos 5x + 1 = 0
NCERT Exemplar -Trigonometric Functions Notes | EduRev
⇒ cos 5x + cos x = 0
NCERT Exemplar -Trigonometric Functions Notes | EduRev
⇒ cos 3x . cos 2x = 0
⇒ cos 3x = 0 or cos 2x = 0
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
Hence, the values of x are NCERT Exemplar -Trigonometric Functions Notes | EduRev

Long Answer Type
Q.20. If sin (θ + α) = a and sin (θ + β) = b, then prove that cos 2(α – β) – 4ab cos (α – β) = 1 – 2a2 – 2b2  
[Hint: Express cos (α – β) = cos ((θ + α) – (θ + β))]
Ans.
Given that:
sin (θ + α) = a and sin  + β) = b  .... (i)
cos  - β) = cos  + α - θ  β) = cos [(θ + α)   + β)] 
∴ cos  - β)= cos  + α) cos  + β) + sin  + α) sin  + β)   
NCERT Exemplar -Trigonometric Functions Notes | EduRev
Now cos 2(α – β) – 4ab cos (α – β)
= 2 cos2 (α – β) – 1 – 4ab cos (α – β) [ cos 2θ = 2 cos2 θ – 1]
NCERT Exemplar -Trigonometric Functions Notes | EduRev
= 2a2 b2 + 2 - 2a2 - 2b2 + 2a2 b2 + NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
= 1  2a2  2b2
Hence, cos  2  - β)  4ab cos  - β) = 1  2a2  2b2.
Hence proved.

Q.21. If cos (θ + φ) = m cos (θ – φ), then prove that NCERT Exemplar -Trigonometric Functions Notes | EduRev
[Hint: Express NCERT Exemplar -Trigonometric Functions Notes | EduRev and apply Componendo and Dividendo]
Ans.
Given that:
cos (θ + φ) = m cos (θ – φ)
NCERT Exemplar -Trigonometric Functions Notes | EduRev
Using componendo and dividendo theorem, we get
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev 
Hence proved.

Q.22. Find the value of the expression
NCERT Exemplar -Trigonometric Functions Notes | EduRev
Ans.
Given that:
NCERT Exemplar -Trigonometric Functions Notes | EduRev
= 3[ cos4 α + sin4  + α)]  2[cos6 α + sin6  - α)] = 3[cos4 α + sin4 α]  2[cos6 α + sin6 α]   
= 3[cos4 α + sin4 α + 2 sin2 α cos2 α - 2 sin2 α cos2 α]  2[(cos2 α + sin2 α)3  3 cos2 α sin2 α (cos2 α + sin2 α)]                                 
=  3[(cos2 α + sin2 α)2  2 sin2 α cos2 α]  2[1  3 cos2 α sin2 α]
= 3[1  2 sin2 α cos2 α]  2[1  3 cos2 α sin2 α]
= 3 – 6  sin2 α cos2 α - 2 + 6 cos2 α sin2 α
= 3  2 = 1
Hence, the value of the given expression is 1.

Q.23. If a cos 2θ + b sin 2θ = c has α and β as its roots, then prove that tanα + tan β = NCERT Exemplar -Trigonometric Functions Notes | EduRev
[Hint: Use the identities cos 2θ = NCERT Exemplar -Trigonometric Functions Notes | EduRev
Ans.
Given that: a cos  + b sin  = c..... (i)
NCERT Exemplar -Trigonometric Functions Notes | EduRev     
NCERT Exemplar -Trigonometric Functions Notes | EduRev    
 a – a tan2θ + 2b tanθ = c(1 +  tan2 θ)
⇒ a  a tan2θ + 2b tan θ = c + c tan2 θ
⇒ a  a tan2θ + 2b tan θ  c tan2 θ  c = 0
⇒ - (a + c) tan2 θ + 2b tan θ + (a  c) = 0
 (a + c) tan2 θ  2b tan θ + (c  a) = 0..... (ii)
Since α and β are the roots of this equation
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev Hence proved.

Q.24.  If x = sec φ – tan φ and y = cosec φ + cot φ then show that xy + x – y + 1 = 0
[Hint: Find xy + 1 and then show that x – y = – (xy + 1)]
Ans.
Given that: x = sec φ – tan φ
and y = cosec φ + cot φ
xy + x – y + 1 = 0
L.H.S.   xy + x – y + 1 = (sec φ – tan φ)
(cosec φ  + cot φ) + (sec φ – tan φ) – (cosec  φ + cot φ) + 1
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev R.H.S.
L.H.S. = R.H.S.
Hence proved.

Q.25. If θ lies in the first quadrant and cosθ = NCERT Exemplar -Trigonometric Functions Notes | EduRev then find the value of 
cos (30° + θ) + cos (45° – θ) + cos (120° – θ).
Ans.
Given that: NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
But θ lies in I quadrant.
∴  NCERT Exemplar -Trigonometric Functions Notes | EduRev
Now cos (30° + θ) + cos (45° – θ) + cos (120° – θ)
= cos 30° cos θ – sin 30° sin θ + cos 45° cos θ + sin 45° sin θ + cos 120° cos θ + sin 120° sin θ
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
Hence, the required solution = NCERT Exemplar -Trigonometric Functions Notes | EduRev

Q.26. Find the value of the expression NCERT Exemplar -Trigonometric Functions Notes | EduRev
[Hint: Simplify the expression to NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
Ans.
 NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
Hence, the required value of the expression = NCERT Exemplar -Trigonometric Functions Notes | EduRev

Q.27.Find the general solution of the equation
5cos2θ + 7sin2θ – 6 = 0
Ans.
5 cos2 θ + 7 sin2 θ  6 = 0
⇒ 5 cos2 θ + 7(1  cos2 θ)  6 = 0
⇒ 5 cos2 θ + 7– 7 cos2 θ  6 = 0
 - 2 cos2 θ + 1 = 0
⇒ 2 cos2 θ = 1
NCERT Exemplar -Trigonometric Functions Notes | EduRev
 NCERT Exemplar -Trigonometric Functions Notes | EduRev
∴  NCERT Exemplar -Trigonometric Functions Notes | EduRev
Hence, the general solution of NCERT Exemplar -Trigonometric Functions Notes | EduRev

Q.28. Find the general solution of the equation
sinx – 3sin2x + sin3x = cosx – 3cos2x + cos3x
Ans.
Given that: sin x – 3 sin 2x + sin 3x = cos x – 3 cos 2x + cos 3x
⇒ (sin 3x + sin x) – 3 sin 2x = (cos 3x + cos x) – 3 cos 2x
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
⇒ 2 sin 2x.cos x – 3 sin 2x = 2 cos 2x .cos x – 3 cos 2x
⇒ 2 sin 2x cos x – 2 cos 2x.cos x = 3 sin 2x – 3 cos 2x
⇒ 2 cos x(sin 2x – cos 2x) = 3(sin 2x – cos 2x)
⇒ 2 cos x (sin 2x – cos 2x) – 3(sin 2x – cos 2x) = 0
⇒ (sin 2x – cos 2x) (2 cos x – 3) = 0
⇒ sin 2x – cos 2x = 0 and 2 cos x – 3 ≠0 [∵  -1 ≤ cos x ≤ 1]
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
Hence, the general solution of the equation is
NCERT Exemplar -Trigonometric Functions Notes | EduRev

Q.29. Find the general solution of the equation ( √3 – 1) cosθ + ( √3 + 1) sinθ = 2
[Hint: Put √3 – 1=  r sinα, √3 + 1 = r cosα which gives tanα = tan NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
Ans.
Given that: ( √3 - 1) cos θ + ( √3 + 1) sin θ = 2
Put √3 - 1 = r sin α, √3 + 1 = r cos α
Squaring and adding, we get
r2 = √3 + 1 - 2 √3 + √3 + 1 + 2 √3
r2 = 8 r = ± 2 √2
Now the given equation can be written as
r sin α cos θ + r cos α sin θ = 2
⇒ r (sin α cos θ + cos α sin θ = 2
⇒ r (sin α cos θ + cos α sin θ) = 2
⇒2 √2 sin ( α + θ) = 2
⇒ sin (α + θ) = NCERT Exemplar -Trigonometric Functions Notes | EduRev
 sin (α+ θ) = NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
Now NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
Putting the value of α in equation (i) we get
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
Hence, the general solution of the given equation is
NCERT Exemplar -Trigonometric Functions Notes | EduRev

Objective Type Questions
Q.30. If sin θ + cosec θ = 2, then sin2 θ + cosec2 θ is equal to
(a) 1
(b) 4
(c) 2
(d) None of these
Ans. (c)
Solution.
Given that: sin θ + cosec θ = 2
Squaring both sides, we get
(sin θ +  cosec θ)2 = (2)2
⇒ sin2  θ +  cosec2 θ + 2 sin θ cosec θ = 4
⇒ sin2 θ + cosec2 θ + 2 NCERT Exemplar -Trigonometric Functions Notes | EduRev
⇒ sin2 θ + cosec2 θ + 2 = 4
⇒ sin2 θ + cosec2 θ = 2
Hence, the correct option is (c).

Q.31. If f (x) = cos2 x + sec2 x, then
(a) f (x) < 1
(b) f (x) = 1
(c) 2 < f (x) < 1
(d) f(x) ≥ 2
[Hint: A.M ≥ G.M.]
Ans. (d)
Solution.
Given that: f(x) = cos2 x + sec2 x
We know that AM ≥ GM
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
⇒ f(x)  2
Hence, the correct option is (d)

Q.32. If tan θ = NCERT Exemplar -Trigonometric Functions Notes | EduRev and tan φ =NCERT Exemplar -Trigonometric Functions Notes | EduRev then the value of θ + φ is
(a) π/6
(b) π
(c) 0
(d) π/4
Ans. (d)
Solution.
We know that
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev 
Hence the correct option is (d).

Q.33. Which of the following is not correct?
(a) NCERT Exemplar -Trigonometric Functions Notes | EduRev
(b) cos θ = 1
(c) NCERT Exemplar -Trigonometric Functions Notes | EduRev
(d) tan θ = 20
Ans. (c)
Solution.
sin θ =NCERT Exemplar -Trigonometric Functions Notes | EduRev is correct. ∵1  sin θ  1
So (a) is correct.
cos θ = 1 is correct. ∵ cos  = 1
So (b) is correct.
NCERT Exemplar -Trigonometric Functions Notes | EduRev 
⇒ cos θ = 2 is not correct.
 - 1  cos θ  1
Hence, (c) is not correct.

Q.34. The value of tan 1° tan 2° tan 3° ... tan 89° is
(a) 0
(b) 1
(c) 1/2
(d) Not defined
Ans. (b)
Solution.
Given that: tan 1° tan 2° tan 3° .... tan 89°
= tan 1° tan 2° tan 3° .... tan 45°.tan (90 – 44°).tan (90 – 43°) ...tan (90 – 1°)
= tan 1° cot 1°.tan 2°.cot 2°.tan 3°.cot 3° ... tan 89°.cot 89°
= 1.1.1.1 ... 1.1 = 1
Hence, the correct option is (b).

Q.35. The value of NCERT Exemplar -Trigonometric Functions Notes | EduRev
(a) 1
(b) √3
(c) √3/2
(d) 2
Ans. (c)
Solution.
Given that: NCERT Exemplar -Trigonometric Functions Notes | EduRev
Let θ = 15°   = 30°
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
Hence, the correct option is (c).

Q.36. The value of cos 1° cos 2° cos 3° ... cos 179° is
(a) 1/√2
(b) 0
(c) 1
(d) –1

Ans. (b)
Solution.
Given expression is cos 1°.cos 2°.cos 3° ... cos 179°
⇒cos 1°.cos 2°.cos 3° ... cos 90°.cos 91° ... cos 179°
⇒ 0 [∵ cos 90° = 0]
Hence, the correct option is (b).

Q.37. If tan θ = 3 and θ lies in third quadrant, then the value of sin θ is
(a)NCERT Exemplar -Trigonometric Functions Notes | EduRev 
(b) NCERT Exemplar -Trigonometric Functions Notes | EduRev
(c) NCERT Exemplar -Trigonometric Functions Notes | EduRev
(d) NCERT Exemplar -Trigonometric Functions Notes | EduRev
Ans. (c)
Solution.
tan θ =  3,  θ lies in third quadrant, it is positive.
 NCERT Exemplar -Trigonometric Functions Notes | EduRevwhere θ lies in third quadrant
Hence the correct option is (c).
NCERT Exemplar -Trigonometric Functions Notes | EduRev

Q.38. The value of tan 75° – cot 75° is equal to
(a) 2√3
(b) 2 + √3
(c) 2 − √3
(d) 1
Ans. (a)
Solution.
The given expression is tan 75° – cot 75°
tan 75° – cot 75° = tan 75° – cot (90 – 15°)
= tan 75° – tan 15° = NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
Hence, the correct option is (a).

Q.39. Which of the following is correct?
(a) sin1° > sin 1
(b) sin 1° < sin 1
(c) sin 1° = sin 1
(d) NCERT Exemplar -Trigonometric Functions Notes | EduRev
Ans. (b)
Solution.
We know that if  θ  increases  then the value of  sin θ  also increases
So, sin  < sin 1 NCERT Exemplar -Trigonometric Functions Notes | EduRev
Hence the correct option is (b).

Q.40. If tan α = NCERT Exemplar -Trigonometric Functions Notes | EduRev then α + β is equal to
(a) NCERT Exemplar -Trigonometric Functions Notes | EduRev
(b) NCERT Exemplar -Trigonometric Functions Notes | EduRev
(c) NCERT Exemplar -Trigonometric Functions Notes | EduRev
(d) NCERT Exemplar -Trigonometric Functions Notes | EduRev
Ans. (d)
Solution.
Given that NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
⇒ tan  + β) NCERT Exemplar -Trigonometric Functions Notes | EduRev
Hence, the correct option is (d).

Q.41. The minimum value of 3 cosx + 4 sinx + 8 is

(a) 5
(b) 9
(c) 7
(d) 3
Ans. (d)
Solution.
The given expression is 3 cos x + 4 sin x + 8
Let y = 3 cos x + 4 sin x + 8
⇒ y  8 = 3 cos x  + 4 sin x
Minimum value of y  8 NCERT Exemplar -Trigonometric Functions Notes | EduRev  
 ⇒ y – 8 = NCERT Exemplar -Trigonometric Functions Notes | EduRev
⇒ y = 8 – 5 = 3
So, the minimum value of the given expression is 3.
Hence, the correct option is (d).

Q.42. The value of tan 3A – tan 2A – tan A is equal to
(a) tan 3A tan 2A tan A
(b) – tan 3A tan 2A tan A
(c) tan A tan 2A – tan 2A tan 3A – tan 3A tan A
(d) None of these
Ans. (a)
Solution.
The given expression is tan 3A – tan 2A – tan A
tan 3A = tan (2A + A)
NCERT Exemplar -Trigonometric Functions Notes | EduRev
⇒ tan 3A(1 – tan 2A tan A) = tan 2A + tan A
⇒ tan 3A – tan 3A tan 2A tan A = tan 2A + tan A
⇒ tan 3A – tan 2A – tan A = tan 3A tan 2A tan A
Hence, the correct option is (a).

Q.43.  The value of sin (45° + θ) – cos (45° – θ) is
(a) 2 cosθ
(b) 2 sinθ
(c) 1
(d) 0
Ans. (d)
Solution.
Given expression is (sin 45° + θ)  cos  (45°- θ)
Sin (45° + θ) = sin 45° cos θ + cos 45° sin θ
NCERT Exemplar -Trigonometric Functions Notes | EduRev
Cos (45° - θ) = cos 45° cos θ + sin 45° sin θ
NCERT Exemplar -Trigonometric Functions Notes | EduRev
Sin (45°+ θ - cos (45°- θ)
NCERT Exemplar -Trigonometric Functions Notes | EduRev
= 0. 
Hence, the correct option is (d).

Q.44. The value of NCERT Exemplar -Trigonometric Functions Notes | EduRev
(a) 2 cosθ
(b) 2 sinθ
(c) 1
(d) 0
Ans. (c)
Solution.
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
Hence, the correct option is (c).

Q.45. cos 2θ cos 2φ + sin2 (θ – φ) – sin2 (θ + φ) is equal to
(a) sin 2(θ + φ)
(b) cos 2(θ + φ)
(c) sin 2(θ – φ)
(d) cos 2(θ – φ)
[Hint: Use sin2 A – sin2 B = sin (A + B) sin (A – B)]
Ans. (b)
Solution.
Given that: cos 2θ.cos 2 φ + sin2  - φ)  sin2 + φ)
Cos  cos 2φ + sin2  - φ)  sin2  + φ)
= cos  cos 2φ + sin  - φ + θ + φ).sin  - φ - θ - φ)
[∵ sin2 A   sin2 B =  sin (A +  B).sin  (A  B)]
= cos  cos 2φ + sin 2θ.sin ( - 2φ)
= cos  cos 2φ - sin  sin 2φ   [  sin(φ - θ) = - sin θ]
=    cos 2(θ + φ)  
Hence, the correct option is (b).    

Q.46. The value of cos 12° + cos 84° + cos 156° + cos 132° is  
(a) NCERT Exemplar -Trigonometric Functions Notes | EduRev
(b) 1
(c) NCERT Exemplar -Trigonometric Functions Notes | EduRev
(d) NCERT Exemplar -Trigonometric Functions Notes | EduRev
Ans. (c)
Solution.
The given expression is cos 12° + cos 84° + cos 156° + cos 132° (cos 132° + cos 12°) + (cos 156° + cos 84°)
NCERT Exemplar -Trigonometric Functions Notes | EduRev
= 2 cos 72°.cos 60° + 2 cos 120°.cos 36°
NCERT Exemplar -Trigonometric Functions Notes | EduRev
= cos 72° – cos 36°
= cos (90° – 18°) – cos 36° = sin 18° – cos 36°
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
Hence, the correct option is (c).

Q.47. If tan A = NCERT Exemplar -Trigonometric Functions Notes | EduRev tan B =NCERT Exemplar -Trigonometric Functions Notes | EduRev then tan (2A + B) is equal to
(a) 1
(b) 2
(c) 3
(d) 4
Ans. (c)
Solution.
Given that: tan A = NCERT Exemplar -Trigonometric Functions Notes | EduRev and tan B =NCERT Exemplar -Trigonometric Functions Notes | EduRev 
tan 2A = NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
So, tan 2A = NCERT Exemplar -Trigonometric Functions Notes | EduRev and tan B =NCERT Exemplar -Trigonometric Functions Notes | EduRev 
tan (2A + B) =NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
Hence, the correct option is (c).

Q.48. The value of NCERT Exemplar -Trigonometric Functions Notes | EduRev is
(a) NCERT Exemplar -Trigonometric Functions Notes | EduRev
(b) NCERT Exemplar -Trigonometric Functions Notes | EduRev
(c) NCERT Exemplar -Trigonometric Functions Notes | EduRev
(d) 1
Ans. (c)
Solution.
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
= – sin 18°.sin 54°
= – sin 18°.sin (90° – 36°) = – sin 18°.cos 36°
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
Hence, the correct option is (c).

Q.49. The value of sin 50° – sin 70° + sin 10° is equal to
(a) 1
(b) 0
(c) 1/2
(d) 2
Ans. (b)
Solution.
Given expression is sin 50° – sin 70° + sin 10°
(sin 50° – sin 70°) + sin 10°= NCERT Exemplar -Trigonometric Functions Notes | EduRev
= 2 cos 60°.(– sin 10°) + sin 10°
NCERT Exemplar -Trigonometric Functions Notes | EduRev
= – sin 10° + sin 10°
= 0
Hence, the correct option is (b).

Q.50. If sin θ + cos θ = 1, then the value of sin 2θ is equal to

(a) 1
(b) 1/2
(c) 0
(d) –1
Ans. (c)
Solution.
Given that:
sin θ + cos θ = 1
Squaring both sides, we get,
⇒ (sin θ + cos θ)2 = (1)2
⇒ sin2 θ + cos2 θ + 2 sin θ cos θ = 1
⇒ 1 + sinθ = 1
⇒ sinθ = 1  1 = 0
Hence, the correct option is (c).


Q.51. If α + β =NCERT Exemplar -Trigonometric Functions Notes | EduRev then the value of (1 + tan α) (1 + tan β) is
(a) 1
(b) 2
(c) – 2
(d) Not defined
Ans. (b)
Solution.
 Given that: NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
⇒ tan α + tan β = 1– tan α tan β
⇒ tan α + tan β + tan α tan β = 1
On adding 1 both sides, we get,
⇒ 1 + tan α + tan β + tan α tan β = 1 + 1
⇒ 1(1 + tan α) + tan β (1 + tan α) = 2
⇒ (1 + tan α )(1 + tan β) = 2
Hence, the correct option is (b)

Q.52. If sin θ = NCERT Exemplar -Trigonometric Functions Notes | EduRev and θ lies in third quadrant then the value of 

NCERT Exemplar -Trigonometric Functions Notes | EduRev is
(a) NCERT Exemplar -Trigonometric Functions Notes | EduRev
(b) NCERT Exemplar -Trigonometric Functions Notes | EduRev
(c) NCERT Exemplar -Trigonometric Functions Notes | EduRev
(d) NCERT Exemplar -Trigonometric Functions Notes | EduRev
Ans. (c)
Solution.
Given that: NCERT Exemplar -Trigonometric Functions Notes | EduRev lies in third quadrant
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev lies in third quadrant
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
Hence, the correct option is (c).

Q.53. Number of solutions of the equation tan x + sec x = 2 cosx lying in the interval [0, 2π] is
(a) 0
(b) 1
(c) 2
(d) 3

Ans. (c)
Solution.
Given equation is tan x + sec x = 2 cos x
NCERT Exemplar -Trigonometric Functions Notes | EduRev
⇒ 1 + sin x = 2 cos2 x
⇒ 2 cos2 x – sin x – 1 = 0
    2(1 – sin2 x) – sin x – 1 = 0
⇒  2 – 2 sin2 x – sin x – 1 = 0
⇒ – 2 sin2 x – sin x + 1 = 0
⇒  sin2 x + sin x – 1 = 0
Since, the equation is a quadratic equation in sin x.
So it will have 2 solutions.
Hence, the correct option is (c).

Q.54. The value of NCERT Exemplar -Trigonometric Functions Notes | EduRevis given by
(a) NCERT Exemplar -Trigonometric Functions Notes | EduRev
(b) 1
(c) NCERT Exemplar -Trigonometric Functions Notes | EduRev
(d) NCERT Exemplar -Trigonometric Functions Notes | EduRev
Ans. (a) 
Solution.
The given expression is NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
Hence, the correct option is (a).

Q.55. If A lies in the second quadrant and 3 tan A + 4 = 0, then the value of 2 cot A  –  5 cos A + sin A is equal to
(a) NCERT Exemplar -Trigonometric Functions Notes | EduRev
(b) NCERT Exemplar -Trigonometric Functions Notes | EduRev
(c) NCERT Exemplar -Trigonometric Functions Notes | EduRev
(d) NCERT Exemplar -Trigonometric Functions Notes | EduRev
Ans. (b)
Solution.
 Given that: 3 tan A + 4 = 0, A lies in second quadrant
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
[A lies in second quadrant]
NCERT Exemplar -Trigonometric Functions Notes | EduRev
and NCERT Exemplar -Trigonometric Functions Notes | EduRev
∴ 2 cot A – 5 cos A + sin A
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
Hence, the correct option is (b).

Q.56. The value of cos2 48° – sin2 12° is
(a) NCERT Exemplar -Trigonometric Functions Notes | EduRev
(b) NCERT Exemplar -Trigonometric Functions Notes | EduRev
(c) NCERT Exemplar -Trigonometric Functions Notes | EduRev
(d) NCERT Exemplar -Trigonometric Functions Notes | EduRev
[Hint:  Use cos2 A – sin2 B = cos (A + B) cos (A – B)]
Ans. (a)
Solution.
Given expression is cos48° – sin2 12°
cos2 48° – sin2 12° = cos (48° + 12°).cos (48° – 12°)
[∵ cos2 A – sin2 B = cos (A + B).cos (A – B)]
= cos 60°.cos 36°
NCERT Exemplar -Trigonometric Functions Notes | EduRev
Hence, the correct option is (a).

Q.57. If tan α =NCERT Exemplar -Trigonometric Functions Notes | EduRev  tan β =NCERT Exemplar -Trigonometric Functions Notes | EduRev then cos 2α is equal to
(a) sin 2β
(b) sin 4β
(c) sin 3β
(d) cos 2β
Ans. (b)
Solution.
Given that: NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
Now NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
cos 2α = sin 4β =NCERT Exemplar -Trigonometric Functions Notes | EduRev
Hence, the correct option is (b).

Q.58. If tan θ = NCERT Exemplar -Trigonometric Functions Notes | EduRev  then b cos 2θ  + a sin 2θ  is equal to
(a) a
(b) b
(c) a/b

(d) None
Ans. (b)
Solution.
Given that: NCERT Exemplar -Trigonometric Functions Notes | EduRev
b cos  +  a sin NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
Hence, the correct option is (b).

Q.59. If for real values of x, cos θ =NCERT Exemplar -Trigonometric Functions Notes | EduRev then
(a) θ is an acute angle
(b) θ is right angle
(c) θ is an obtuse angle
(d) No value of θ is possible
Ans. (d)
Solution.
Given that: NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
⇒ x2 + 1 =  x cos θ
 x2  x cos θ + 1 = 0
For real value of x, b2  4ac  0
⇒ (- cos θ)2  4 × 1 × 1  0
⇒ cos2 θ  4  0
 cos2 θ  4
⇒ cos θ  2 [- 1  cos θ  1]
So, the value of θ is not possible.
Hence, the correct options (d).

Fill in the blanks

Q.60. The value of NCERT Exemplar -Trigonometric Functions Notes | EduRevis _______ .
Ans. 
NCERT Exemplar -Trigonometric Functions Notes | EduRev
Hence, the value of filler is 1.

Q.61. If k =NCERT Exemplar -Trigonometric Functions Notes | EduRev then the numerical value of k is _______.
Ans. 
Given that: k = NCERT Exemplar -Trigonometric Functions Notes | EduRev
⇒ k = sin 10°. Sin 50°. Sin 70°
⇒ k = sin 10° sin (90° -  40°) sin (90° - 20°)
⇒ k = sin 10° cos 40° cos 20°
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
Hence, the value of the filler is NCERT Exemplar -Trigonometric Functions Notes | EduRev

Q.62. If tan A =NCERT Exemplar -Trigonometric Functions Notes | EduRev, then tan 2A = _______.
Ans.
Given that:  tan A = NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
So, tan 2A = tan B
Hence, the value of the filler is tan B.

Q.63. If sin x + cos x = a, then
(i) sin6 x + cos6 x = _______
(ii) | sin x – cos x | = _______.
Ans. 
Given that: sin x + cos x = a
Squaring both sides, we get,
(sin x + cos x)2 = a2
⇒ sin2 x + cos2 x + 2 sin x cos x = a2
⇒ 1 + 2 sin x cos x = a2
⇒ sin x cos x = NCERT Exemplar -Trigonometric Functions Notes | EduRev ...(i)
(i) sin6 x + cos6 x = (sin2 x)3 + (cos2 x)3
= (sin2 x + cos2 x)3 – 3 sin2 x cos2 x(sin2 x + cos2 x)
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
Hence, the value of the filler is NCERT Exemplar -Trigonometric Functions Notes | EduRev
(ii) |sin x  cos x|2 = sin2 x + cos2 x   2 sin x cos x
NCERT Exemplar -Trigonometric Functions Notes | EduRev
= 1 – (a2 – 1)
= 1 – a2 + 1
= 2  a2
    |sin x  cos x| = NCERT Exemplar -Trigonometric Functions Notes | EduRev   
NCERT Exemplar -Trigonometric Functions Notes | EduRev
Hence, the value of the filler is NCERT Exemplar -Trigonometric Functions Notes | EduRev

Q.64. In a triangle ABC with ∠C = 90° the equation whose roots are tan A and tan B is _______.
[Hint: A + B = 90° ⇒ tan A tan B = 1 and tan A + tan B = NCERT Exemplar -Trigonometric Functions Notes | EduRev
Ans. 
Given a ΔABC with ∠C = 90°
x2    (tan A + tan B)x + tan A.tan B = 0
A + B = 90°  [∵  ∠C = 90°]
⇒ tan (A + B) = tan 90°
NCERT Exemplar -Trigonometric Functions Notes | EduRev
 1 – tan A tan B = 0
 tan A tan B = 1……(i)
Now  tan A + tan B = NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
∴ tan A + tan B =NCERT Exemplar -Trigonometric Functions Notes | EduRev……(ii)
From (i) and (ii) we get
NCERT Exemplar -Trigonometric Functions Notes | EduRev
Hence, the value of the filler is NCERT Exemplar -Trigonometric Functions Notes | EduRev

Q.65. 3 (sin x – cos x)4 + 6 (sin x + cos x)2 + 4 (sin6 x + cos6 x) = ______.
Ans. 
Given expression is 3(sin x – cos x)4 + 6(sin x + cos x)2 + 4(sin6 x + cos6 x)
= 3[sin2 x + cos2 x – 2 sin x cos x]2 + 6(sin2 x + cos2 x + 2 sin x cos x) + 4 [(sin2 x)3 + (cos2 x)3]
= 3[1 – 2 sin x cos x]2 + 6(1 + 2 sin x cos x) + 4[(sin2 x + cos2 x)3 – 3 sinx cos2 x (sin2 x + cos2 x)]
= 3[1 + 4 sinx cos2 x – 4 sin x cos x] + 6(1 + 2 sin x cos x) + 4[1 – 3 sin2 x cos2 x]
= 3 + 12 sin2 x cos2 x – 12 sin x cos x + 6 + 12 sin x cos x + 4 – 12 sin2 x cos2 x
= 3 + 6 + 4 = 13
Hence, the value of the filler is 13.

Q.66. Given x > 0, the values of f (x) = – 3 cos NCERT Exemplar -Trigonometric Functions Notes | EduRev lie in the interval _______.
Ans. 
Given that: f(x) =NCERT Exemplar -Trigonometric Functions Notes | EduRev
Put NCERT Exemplar -Trigonometric Functions Notes | EduRev
 f(x) = - 3 cos y
 - 1  cos y  1
3  - 3 cos y  -3
⇒ - 3  - 3 cos y  3
 - 3  - 3 cos ≤ NCERT Exemplar -Trigonometric Functions Notes | EduRev, x > 0
Hence, the value of the filler is [– 3, 3].

Q.67. The maximum distance of a point on the graph of the function y = √3 sin x + cos x from x-axis is _______.
Ans.
Given that y = √3 sin x + cos x ...(i)
∴ The maximum distance from a point on the graph of eqn. (i) from x-axis
NCERT Exemplar -Trigonometric Functions Notes | EduRev
Hence, the value of the filler is 2.

state whether the statements is True or False? Also give justification.
Q.68. If tan A =NCERT Exemplar -Trigonometric Functions Notes | EduRev then tan 2A = tan B
Ans. 
Given that: tan A NCERT Exemplar -Trigonometric Functions Notes | EduRev
tan 2A = NCERT Exemplar -Trigonometric Functions Notes | EduRev
∴ tan 2A = tan B
Hence, the statement is ‘True’.

Q.69. The equality sin A + sin 2A + sin 3A = 3 holds for some real value of A.
Ans. 
Given that: sin A + sin 2A + sin 3A = 3
Since the maximum value of sin A is 1 but for sin 2A
and sin 3A it is not equal to 1. So it is not possible.
Hence, the given statement is ‘False’.

Q.70. sin 10° is greater than cos 10°.

Ans. 
If sin 10° > cos 10°
⇒ sin 10° > cos (90° – 80°)
⇒sin 10° > sin 80°
which is not possible.
Hence, the statement is ‘False’.

Q.71. NCERT Exemplar -Trigonometric Functions Notes | EduRev
Ans. 
 L.H.S. NCERT Exemplar -Trigonometric Functions Notes | EduRev
= cos 24°.cos 48°.cos 96°.cos 192°
NCERT Exemplar -Trigonometric Functions Notes | EduRev[(2 sin 24 cos 24 )(2 cos 48 )(2 cos 96 )(2 cos 192 )]
NCERT Exemplar -Trigonometric Functions Notes | EduRev[sin 48 .2 cos 48 (2 cos 96 )(2 cos 192 )]
NCERT Exemplar -Trigonometric Functions Notes | EduRev[2 sin 48 cos 48 (2 cos 96 )(2 cos 192 )]
NCERT Exemplar -Trigonometric Functions Notes | EduRev[sin 96 (2 cos 96 )(2 cos 192 )]
NCERT Exemplar -Trigonometric Functions Notes | EduRev[2 sin 96 . cos 96 (2 cos 192 )]
NCERT Exemplar -Trigonometric Functions Notes | EduRev [sin 192 .(2 cos 192 )]
NCERT Exemplar -Trigonometric Functions Notes | EduRev2 sin 192 cos 192
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev [∵ sin (360° + θ) = sin θ]
NCERT Exemplar -Trigonometric Functions Notes | EduRev R.H.S.
Hence, the given statement is ‘True’.

Q.72. One value of θ which satisfies the equation sin4 θ – 2sin2 θ – 1 lies between 0 and 2π.
Ans. Given equation is
sin4 θ – 2 sin2 θ – 1 = 0
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
= 1 ± √2
∴ sin2 θ = (1 + √2 ) or (1 - √2 )
⇒ - 1 ≤ sin q ≤ 1
⇒ sin2 θ ≤  1 but sin2 θ = (1 + √2 ) or (1 - √2 )
Which is not possible.
Hence, the given statement is ‘False’.

Q.73. If cosec x = 1 + cot x then x = 2nπ, 2nπ +NCERT Exemplar -Trigonometric Functions Notes | EduRev
Ans. 
Given that: cosec x = 1 + cot x
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
⇒ sin x + cos x = 1
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
or NCERT Exemplar -Trigonometric Functions Notes | EduRev
Hence, the given statement is ‘True’.

Q.74. If tan θ + tan 2θ + √3 tan θ tan 2θ = √3 , then NCERT Exemplar -Trigonometric Functions Notes | EduRev
Ans.
Given that:tan θ + tan 2θ + √3 tan θ tan 2θ = √3
⇒ tan θ + tan 2θ = √3 - √3 tan θ tan 2θ
⇒ tan θ + tan 2θ = √3 (1 - tan θ tan 2θ)
NCERT Exemplar -Trigonometric Functions Notes | EduRev
⇒ tan (θ + 2θ) = √3
NCERT Exemplar -Trigonometric Functions Notes | EduRev
So NCERT Exemplar -Trigonometric Functions Notes | EduRev
Hence, the given statement is ‘True’.

Q.75. If tan (π cosθ) = cot (π sinθ), thenNCERT Exemplar -Trigonometric Functions Notes | EduRev
Ans.
Given that: tan (π cos θ) = cot (π sin θ)
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
Hence, the given statement is ‘True’.

Q.76. In the following match each item given under the column C1 to its correct answer given under the column C2 :

(a) sin (x + y) sin (x – y)
(i) cos2 x – sin2 y
(b) cos (x + y) cos (x – y) 
NCERT Exemplar -Trigonometric Functions Notes | EduRev 
NCERT Exemplar -Trigonometric Functions Notes | EduRev 
NCERT Exemplar -Trigonometric Functions Notes | EduRev 
NCERT Exemplar -Trigonometric Functions Notes | EduRev 
 (iv) sin2 x – sin2 y

Ans. 
(a) sin (x + y) sin (x  y) = sin2 x  sin2 y
 (a)↔(iv)
(b) cos (x + y) cos (x  y) = cos2 x  cos2 y
 (b)  (i)
(c) NCERT Exemplar -Trigonometric Functions Notes | EduRev
NCERT Exemplar -Trigonometric Functions Notes | EduRev
 (c)  (ii)
(d) NCERT Exemplar -Trigonometric Functions Notes | EduRev
 (d)  (iii)
Hence,
(a)  (iv)
(b) ↔ (i)
(c) ↔ (ii)
(d) ↔ (iii).

Offer running on EduRev: Apply code STAYHOME200 to get INR 200 off on our premium plan EduRev Infinity!

Complete Syllabus of JEE

JEE

Dynamic Test

Content Category

Related Searches

Objective type Questions

,

Extra Questions

,

practice quizzes

,

Semester Notes

,

Previous Year Questions with Solutions

,

mock tests for examination

,

Sample Paper

,

past year papers

,

shortcuts and tricks

,

NCERT Exemplar -Trigonometric Functions Notes | EduRev

,

Exam

,

study material

,

NCERT Exemplar -Trigonometric Functions Notes | EduRev

,

Viva Questions

,

NCERT Exemplar -Trigonometric Functions Notes | EduRev

,

MCQs

,

Important questions

,

Free

,

ppt

,

pdf

,

video lectures

,

Summary

;