**NCERT Solutions for Class 12 Math Chapter 2 – Inverse Trigonometric Functions**

**Chapter 2 – Inverse Trigonometric Functions**

#### Question 1:

Find the principal value of

#### Answer:

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

#### Answer:

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)

#### Answer:

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

#### Answer:

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

#### Answer:

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)

#### Answer:

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

#### Page No 42:

#### Question 7:

Find the principal value of

#### Answer:

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

#### Answer:

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

#### Answer:

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

#### Answer:

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

#### Answer:

#### Question 12:

Find the value of

#### Answer:

#### Question 13:

Find the value of if sin^{−1} *x *= *y*, then

**(****A)** **(B)**

**(****C)** **(D) **

#### Answer:

It is given that sin^{−1} *x *= *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) **

#### Answer:

#### Page No 47:

#### Question 1:

Prove

#### Answer:

To prove:

Let *x* = sin*θ*. Then,

We have,

R.H.S. =

= 3*θ*

= L.H.S.

#### Question 2:

Prove

#### Answer:

To prove:

Let *x* = cos*θ*. Then, cos^{−1} *x* =*θ*.

We have,

#### Question 3:

Prove

#### Answer:

To prove:

#### Question 4:

Prove

#### Answer:

To prove:

#### Question 5:

Write the function in the simplest form:

#### Answer:

#### Question 6:

Write the function in the simplest form:

#### Answer:

Put *x* = cosec *θ* ⇒ *θ* = cosec^{−1} *x*

#### Question 7:

Write the function in the simplest form:

#### Answer:

#### Question 8:

Write the function in the simplest form:

#### Answer:

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

#### Page No 48:

#### Question 9:

Write the function in the simplest form:

#### Answer:

#### Question 10:

Write the function in the simplest form:

#### Answer:

#### Question 11:

Find the value of

#### Answer:

Let. Then,

#### Question 12:

Find the value of

#### Answer:

#### Question 13:

Find the value of

#### Answer:

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*.

#### Answer:

On squaring both sides, we get:

Hence, the value of *x* is

#### Question 15:

If, then find the value of *x*.

#### Answer:

Hence, the value of *x* is

#### Question 16:

Find the values of

#### Answer:

We know that sin^{−1} (sin *x*) =* x* if, which is the principal value branch of sin^{−1}*x*.

Here,

Now, can be written as:

#### Question 17:

Find the values of

#### Answer:

We know that tan^{−1} (tan *x*) =* x* if, which is the principal value branch of tan^{−1}*x*.

Here,

Now, can be written as:

#### Question 18:

Find the values of

#### Answer:

Let. Then,

#### Question 19:

Find the values of is equal to

**(A)** **(B)** **(C)** **(D)**

#### Answer:

We know that cos^{−1} (cos *x*) =* x* if, which is the principal value branch of cos ^{−1}*x*.

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

The correct answer is B.

#### Question 20:

Find the values of is equal to

**(A)** **(B)** **(C)** **(D)** 1

#### Answer:

Let. Then,

We know that the range of the principal value branch of.

∴

The correct answer is D.

#### Question 21:

Find the values of is equal to

**(A)** π **(B)** **(C)** 0 **(D)**

#### Answer:

Let. Then,

We know that the range of the principal value branch of

Let.

The range of the principal value branch of

The correct answer is B.

#### Page No 51:

#### Question 1:

Find the value of

#### Answer:

We know that cos^{−1} (cos *x*) =* x* if, which is the principal value branch of cos ^{−1}*x*.

Here,

Now, can be written as:

#### Question 2:

Find the value of

#### Answer:

We know that tan^{−1} (tan *x*) =* x* if, which is the principal value branch of tan ^{−1}*x*.

Here,

Now,

can be written as:

#### Question 3:

Prove

#### Answer:

Now, we have:

#### Question 4:

Prove

#### Answer:

Now, we have:

#### Question 5:

Prove

#### Answer:

Now, we will prove that:

#### Question 6:

Prove

#### Answer:

Now, we have:

#### Question 7:

Prove

#### Answer:

Using (1) and (2), we have

#### Question 8:

Prove

#### Answer:

#### Page No 52:

#### Question 9:

Prove

#### Answer:

#### Question 10:

Prove

#### Answer:

#### Question 11:

Prove [**Hint: **put*x* = cos 2*θ*]

#### Answer:

#### Question 12:

Prove

#### Answer:

#### Question 13:

Solve

#### Answer:

#### Question 14:

Solve

#### Answer:

#### Question 15:

Solveis equal to

**(A) ** (**B) ** (**C) ** (**D) **

#### Answer:

Let tan^{−1} *x* = *y*. Then,

The correct answer is D.

#### Question 16:

Solve**, **then *x* is equal to

**(****A) ** (**B) ** (**C)** 0 (**D) **

#### Answer:

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:

Solveis equal to

**(A)** **(B).** **(C)** **(D) **

#### Answer:

Hence, the correct answer is **C**.