Page 1
1. co n a b m s si si s n, a b n co
2 2 2 2
m n a b If and prove that,
Sol:
We have
=
+
=
=
=
=
Hence,
2. n x a s e c bta n , y a b s e c ta If and prove that
2 2 2 2
x y a b .
Sol:
We have
=
Page 2
1. co n a b m s si si s n, a b n co
2 2 2 2
m n a b If and prove that,
Sol:
We have
=
+
=
=
=
=
Hence,
2. n x a s e c bta n , y a b s e c ta If and prove that
2 2 2 2
x y a b .
Sol:
We have
=

=
=
=
=
Hence,
3. If sin cos 1
x y
a b
and cos sin 1,
x y
a b
prove that
2 2
2 2
2.
x y
a b
Sol:
We have
Squaring both side, we have:
Again,
Now, adding (i) and (ii), we get:
4. If sec tan mand sec tan , n show that 1. m n
Sol:
We have
Again,
Now, multiplying (i) and (ii), we get:
Page 3
1. co n a b m s si si s n, a b n co
2 2 2 2
m n a b If and prove that,
Sol:
We have
=
+
=
=
=
=
Hence,
2. n x a s e c bta n , y a b s e c ta If and prove that
2 2 2 2
x y a b .
Sol:
We have
=

=
=
=
=
Hence,
3. If sin cos 1
x y
a b
and cos sin 1,
x y
a b
prove that
2 2
2 2
2.
x y
a b
Sol:
We have
Squaring both side, we have:
Again,
Now, adding (i) and (ii), we get:
4. If sec tan mand sec tan , n show that 1. m n
Sol:
We have
Again,
Now, multiplying (i) and (ii), we get:
=
=
=1
5. If cos cot e c mand sec cot , c o n show that 1. m n
Sol:
We have
=
=
6. If
3
cos x a and
3
sin , y b prove that
2 2
3 3
1.
x y
a b
Sol:
=
=
=
=
= 1
7. If tan sin mand tan sin , n prove that
2
2 2
16 . m n m n
Sol:
=
=
=
=
=
Page 4
1. co n a b m s si si s n, a b n co
2 2 2 2
m n a b If and prove that,
Sol:
We have
=
+
=
=
=
=
Hence,
2. n x a s e c bta n , y a b s e c ta If and prove that
2 2 2 2
x y a b .
Sol:
We have
=

=
=
=
=
Hence,
3. If sin cos 1
x y
a b
and cos sin 1,
x y
a b
prove that
2 2
2 2
2.
x y
a b
Sol:
We have
Squaring both side, we have:
Again,
Now, adding (i) and (ii), we get:
4. If sec tan mand sec tan , n show that 1. m n
Sol:
We have
Again,
Now, multiplying (i) and (ii), we get:
=
=
=1
5. If cos cot e c mand sec cot , c o n show that 1. m n
Sol:
We have
=
=
6. If
3
cos x a and
3
sin , y b prove that
2 2
3 3
1.
x y
a b
Sol:
=
=
=
=
= 1
7. If tan sin mand tan sin , n prove that
2
2 2
16 . m n m n
Sol:
=
=
=
=
=
=
=
=
=
= s
=
=
=
8. If cot tan mand sec cos n prove that
2 2
2 2
3 3
( ) ( ) 1 m n m n
Sol:
Now,
=
=
=
=
=
=
=
=
=
=
=
=
=
Page 5
1. co n a b m s si si s n, a b n co
2 2 2 2
m n a b If and prove that,
Sol:
We have
=
+
=
=
=
=
Hence,
2. n x a s e c bta n , y a b s e c ta If and prove that
2 2 2 2
x y a b .
Sol:
We have
=

=
=
=
=
Hence,
3. If sin cos 1
x y
a b
and cos sin 1,
x y
a b
prove that
2 2
2 2
2.
x y
a b
Sol:
We have
Squaring both side, we have:
Again,
Now, adding (i) and (ii), we get:
4. If sec tan mand sec tan , n show that 1. m n
Sol:
We have
Again,
Now, multiplying (i) and (ii), we get:
=
=
=1
5. If cos cot e c mand sec cot , c o n show that 1. m n
Sol:
We have
=
=
6. If
3
cos x a and
3
sin , y b prove that
2 2
3 3
1.
x y
a b
Sol:
=
=
=
=
= 1
7. If tan sin mand tan sin , n prove that
2
2 2
16 . m n m n
Sol:
=
=
=
=
=
=
=
=
=
= s
=
=
=
8. If cot tan mand sec cos n prove that
2 2
2 2
3 3
( ) ( ) 1 m n m n
Sol:
Now,
=
=
=
=
=
=
=
=
=
=
=
=
=
=
= RHS
Hence proved.
9. If
3 3
(cos sin ) (sec cos ) , e c a a n d b prove that
2 2 2 2
( ) 1 a b a b
Sol:
=
=
2
3
1
3
cos
a=
sin
=
=
=
=
=
=
=
=
= RHS
Hence, proved.
10. If 2sin 3cos 2, prove that 3sin 2cos 3.
Sol:
=
=
= 4+9
= 13
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