## EXERCISE 2.3

#### 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**.