# NCERT Solutions Class 12 Maths: Chapter-2 Inverse Trigonometric Functions

**Q1 : Find the principal value of **

**Answer :**

Let =y. Then sin y=

We know that the range of the principal value branch of sin^{-1} is

and sin

Therefore, the principal value of

**Q2 : ****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.

**Q3 : ****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

**Q4 : ****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

**Q5 : ****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

**Q6 : 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

**Q7 : 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

**Q8 :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

**Q9 : 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

**Q10 : 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

**Q11 :****Find the value of**

** Answer : **

**Q12 :Find the value ofAnswer :**

**Q13 :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,.

**Q14 :Find the value of is equal to **

**(A) π (B) (C) (D)**

**Answer**

**Exercise 2.2 : Solutions of Questions on Page Number : 47**

**Q1 :Prove **

** Answer :**

To prove:

Let x = sinθ. Then,

We have,

R.H.S. =

= 3θ

= L.H.S.

**Q2 :Prove**

** Answer :**

To prove:

Let x = cosθ. Then, cos^{-1} x =θ.

We have,

**Q3 :Prove **

** Answer :**

To prove:

**Q4 :Prove **

** Answer :**

To prove:

**Q6 :Write the function in the simplest form: **

**Answer :**

Put x = cosec θ ⇒ θ = cosec

^{-1}x

**Q7 :Write the function in the simplest form:**

** Answer :**

**Q8 :Write the function in the simplest form: **

**Answer :**

**Q9 :Write the function in the simplest form: **

**Answer :**

**Q10 :Write the function in the simplest form: **

**Answer :**

**Q11 :Find the value of**

** Answer :**

Let. Then,

**Q12 :Find the value of **

** Answer : **

**Q13 :Find the value of **

**Answer :**

Let x = tan θ. Then, θ = tan

^{-1}x.

Let y = tan Φ. Then, Φ = tan

^{-1}y.

**Q14 :If, then find the value of x.**

** Answer : **

On squaring both sides, we get:

Hence, the value of x is

**Q15 :If, then find the value of x.**

** Answer : **

Hence, the value of x is

**Q16 :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:

**Q17 :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:

**Q18 :****Find the values of **

**Answer :**

Let. Then,

**Q19 :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:

The correct answer is B.

**Q20 :Find the values of is equal to **

**Answer :**

Let . Then

We know that the range of the principle value branch of

The correct answer is D.

**Q21 :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.

**Exercise Miscellaneous : Solutions of Questions on Page Number : 51**

**Q1 :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:

**Q2 :****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:

**Q3 :Prove**

** Answer : **

Now, we have:

**Q4 :Prove **

**Answer :**

Now, we have:

**Q5 :Prove**

** Answer : **

Now, we will prove that:

**Q6 :Prove**

** Answer : **

Now, we have:

**Q7 :Prove**

** Answer : **

Using (1) and (2), we have

**Q8 :Prove **

**Answer :**

**Q9 :Prove**

** Answer : **

**Q10 :Prove**

** Answer : **

**Q11 :Prove [Hint: putx = cos 2θ]**

** Answer : **

**Q12 :Prove**

** Answer : **

**Q13 :Solve**

** Answer : **

**Q14: Solve **

**Answer:**

**Q15 :Solveis equal to**

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

** Answer :**

Let tan – 1 x = y. Then,

The correct answer is D.

**Q16 :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.

**Q17 :Solve is equal to**

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

** Answer :**

Hence, the correct answer is C.