# NCERT Solutions For Class 12 Maths Chapter 2 – Inverse Trigonometric Functions

## NCERT Solutions Of Class 12 Maths Chapter 2  – Inverse Trigonometric Functions

 Name OF The Section Topic Name 2 Inverse Trigonometric Functions 2.1 Introduction 2.2 Basic Concepts 2.3 Properties of Inverse Trigonometric Functions

### Class 12 Maths Chapter 2 NCERT Solutions – 2.1 Introduction

Question 1:
Find the principal value of
Let sin-1 Then sin y =

We know that the range of the principal value branch of sin−1 is
and sin
Therefore, the principal value of

Question 2:
Find the principal value of

We know that the range of the principal value branch of cos−1 is

Therefore, the principal value of

Question 3:

Find the principal value of cosec−1 (2)
Let cosec−1 (2) = y. Then,
We know that the range of the principal value branch of cosec−1 is

Therefore, the principal value of

Question 4:
Find the principal value of

We know that the range of the principal value branch of tan−1 is

Therefore, the principal value of

Question 5:
Find the principal value of

We know that the range of the principal value branch of cos−1 is

Therefore, the principal value of

Question 6:

Find the principal value of tan−1 (−1)
Let tan−1 (−1) = y. Then,

We know that the range of the principal value branch of tan−1 is

Therefore, the principal value of

Question 7:
Find the principal value of

We know that the range of the principal value branch of sec−1 is

Therefore, the principal value of

Question 8:
Find the principal value of

We know that the range of the principal value branch of cot−1 is (0,π) and

Therefore, the principal value of

Question 9:
Find the principal value of

We know that the range of the principal value branch of cos−1 is [0,π] and

Therefore, the principal value of

Question 10:
Find the principal value of

We know that the range of the principal value branch of cosec−1 is

Therefore, the principal value of

Question 11:
Find the value of

Question 12:
Find the value of

Question 13:

Find the value of if sin−1 y, then
(A) (B)
(C) (D)

It is given that sin−1 y.

We know that the range of the principal value branch of sin−1 is

Therefore,

Question 14:
Find the value of is equal to
(A) π (B) (C) (D)

### NCERT Solutions For Class 12 Maths Chapter 2 Inverse Trigonometric Functions – Exercise 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,

Question 3:
Prove
To prove:

Question 4:
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:

Dividing numerator and denominator by cos 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.

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

Let

The range of the principal value branch of

### NCERT Solutions Class 12 Maths Chapter 2 Inverse Trigonometric Functions – Miscellaneous Solutions

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

Question 2:
Find the value 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 3:
Prove

Now, we have:

Question 4:
Prove

Now, we have:

Question 5:
Prove

Now, we will prove that:

Question 6:
Prove

Now, we have:

Question 7:
Prove

Using (1) and (2), we have

Question 8:
Prove

Question 9:
Prove

Question 10:
Prove

Question 11:
Prove [Hint: put x = cos 2θ] Answer:

Question 12:
Prove

Question 13:
Solve

Question 14:
Solve

Question 15:
Solve is equal to
(A) (B) (C) (D)
Let tan−1x = y. Then,

Question 16:
Solve , then x is equal to
(A) (B) (C) 0 (D)

Therefore, from equation (1), we have

Put x = sin y. Then, we have:

But, when

it can be observed that:

is not the solution of the given equation.

Thus, x = 0.

Hence, the correct answer is C.

Question 17:
Solve is equal to
(A) (B) (C) (D)