## NCERT Solutions for Class 12 Maths Chapter 2 – Inverse Trigonometric Functions Ex 2.2

#### Question 1:

Prove

To prove:
Let x = sinθ. Then,
We have,
R.H.S. =
= 3θ
= L.H.S.

#### Question 2:

Prove

To prove:
Let x = cosθ. Then, cos−1 x =θ.
We have,

Prove

To prove:

Prove

To prove:

#### Question 5:

Write the function in the simplest form:

#### Question 6:

Write the function in the simplest form:

Put x = cosec θ ⇒ θ = cosec−1 x

#### Question 7:

Write the function in the simplest form:

#### Question 8:

Write the function in the simplest form:

tan-1cosx-sinxcosx+sinx=tan-11-sinxcosx1+sinxcosx=tan-11-tanx1+tanx=tan-11-tan-1tanx        tan-1x-y1+xy=tan-1x-tan-1y=π4-x

#### Question 9:

Write the function in the simplest form:

#### Question 10:

Write the function in the simplest form:

#### Question 11:

Find the value of

Let. Then,

#### Question 12:

Find the value of

#### Question 13:

Find the value of

Let x = tan θ. Then, θ = tan−1 x.
Let y = tan Φ. Then, Φ = tan−1 y.

#### Question 14:

If, then find the value of x.

On squaring both sides, we get:
Hence, the value of x is

#### Question 15:

If, then find the value of x.

Hence, the value of x is

#### Question 16:

Find the values of

We know that sin−1 (sin x) = x if, which is the principal value branch of sin−1x.
Here,
Now, can be written as:

#### Question 17:

Find the values of

We know that tan−1 (tan x) = x if, which is the principal value branch of tan−1x.
Here,
Now, can be written as:

#### Question 18:

Find the values of

Let. Then,

#### Question 19:

Find the values of is equal to
(A)  (B)  (C)  (D)

We know that cos−1 (cos x) = x if, which is the principal value branch of cos −1x.
Here,
Now, can be written as:

cos-1cos7π6 = cos-1cosπ+π6cos-1cos7π6 = cos-1- cosπ6             as, cosπ+θ = – cos θcos-1cos7π6  = cos-1- cosπ-5π6cos-1cos7π6 = cos-1– cos 5π6   as, cosπ-θ = – cos θ

#### Question 20:

Find the values of is equal to
(A)  (B)  (C)  (D) 1

Let. Then,
We know that the range of the principal value branch of.

#### Question 21:

Find the values of is equal to
(A) π (B)  (C) 0 (D)