Class 12 NCERT Solutions- Mathematics Part I – Chapter 2 Inverse Trigonometric Functions – Miscellaneous Exercise on Chapter 2

Question 1. Find the value of 

Solution: 

We know that 

Here, 

Now,  can be written as :

, where 

Hence, the value of  = π/6

Question 2. Find the value of 

Solution: 

We know that 

Here, 

Now,  can be written as:

 where 

Hence, the value of  = π/6

Question 3. Prove 

Solution: 

Let   -(1)

sin x = 3/5

So,= 4/5

tan x = 3/4

Hence, 

Now put the value of x from eq(1), we get

Now, we have

L.H.S 

=                 –

Hence, proved.

Question 4. Prove 

Solution: 

Let

Then sin x = 8/17 

 cos x = = 15/17

Therefore, 

         -(1)

Now, let 

Then, sin y = 3/5 

 = 4/5

     -(2)

Now, we have:

L.H.S.

 From equation(1) and (2), we get

= 

= 

=                                  –

= 

Hence proved

Question 5. Prove 

Solution: 

Let 

Then, cos x = 4/5

 = 3/5

        -(1)

Now let 

Then, cos y = 3/4 

            -(2)

Let 

Then, cos z = 33/65

sin z = 56/65 

    -(3)

Now, we will prove that : 

L.H.S. 

From equation (1) and equation (2)

= 

=          –

= 

= 

Using equation(3)

= 

Hence proved

Question 6. Prove 

Solution: 

Let

Then, sin x = 3/5 

 = 4/5

           -(1)

Now, let 

Then, cos y = 12/13 and sin y = 5/13

        -(2)                                                             

Let 

Then, sin z = 56/65 and cos z = 33/65

           -(3)

Now, we have:

L.H.S.=

From equation(1) and equation(2)

=

=                –

= 

= 

From equation (3)

= 

Hence proved                                                                                                           

Question 7. Prove 

Solution:

Let

Then, sin x = 5/13 and cos x = 12/13.

                   -(1)

Let 

Then, cos y = 3/5 and sin y = 4/5

                -(2)

From equation(1) and (2), we have

R.H.S.

=

=           –

=

=

L.H.S = R.H.S

Hence proved

Question 9. Prove 

Solution:

Let x = tan²θ

Then,

Now, we have

R.H.S = 

L.H.S = R.H.S

Hence proved

Question 9. Prove 

Solution:

Consider

By rationalizing

=

=

=

= 

L.H.S = = x/2

L.H.S = R.H.S

Hence proved

Question 10. Prove 

Solution:

Put so that, 

Then, we have :

LHS = 

= 

= 

= 

= 

      – 

L.H.S = R.H.S

Hence Proved

Question 11. Solve 

Solution:

=                      –

= 

= 

= cos x/sin x

= cot x =1

Therefore, x = π/4​

Question 12. Solve 

Solution:

Let x = tanθ

π/4 – θ = θ/2

θ = π/6

So, x = tan(π/6) = 1/√3

Question 13. Solve  is equal to

(A)     (B)  (C) (D) 

Solution:

Let tan y = x, 

Let  Then,

So, the correct answer is D.

Question 14. Solve , then x is equal to

(A) 0, 1/2      (B) 1, 1/2      (C) 0     (D) 1/2 

Solution:

        -(1)

Let 

Therefore, from equation(1), we have

Put x = siny then, we have:

sin y = 0 or 1/2

x = 0 or x = 1/2

But, when x = 1/2 it can be observed that:

L.H.S. = 

= 

= 

= 

x = 1/2 is not the solution of given equation.

Thus, x = 0 

Hence, the correct answer is C

Prove 

Solution:

L.H.S.

=      –

= 

= 

= 

= 

= 

= 

= π/4 

L.H.S = R.H.S

Hence proved

Prove 

Solution:

L.H.S. = 

= 

Using 

=       -(1)

Now, let  Then, 

Using equation(1), we get,

= 

L.H.S = R.H.S

Hence Proved

Solve  is equal to

(A) π​/2      (B) π​/3     (C) π​/4      (D) -3π​/4  

Solution

                –

Hence, the correct answer is C