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## 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

**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

**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:**

### NCERT Solutions For Class 12 Maths Chapter 2 Inverse Trigonometric Functions – Exercise 2.2

**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:
**Dividing numerator and denominator by cos x

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*.

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.

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

**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:**

**
Question 9:
**Prove

**Answer:**

**
Question 10:
**Prove

**Answer:**

**
Question 11:
**Prove [

**Hint:**put

*x*= cos 2

*θ*]

**Answer:**

Question 12:

Prove

Question 12:

**Answer:**

**
Question 13:
**Solve

**Answer:**

**
Question 14:
**Solve

**Answer:**

**
Question 15:
**Solve is 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:
**Solve is equal to

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

**Answer:**

Hence, the correct answer is C.