NCERT Solutions for Class 12 Science Maths Chapter 2 – Inverse Trigonometric Functions

Comprehensive solutions for Class 12 Science Mathematics Chapter 2, Inverse Trigonometric Functions, are available here, complete with straightforward step-by-step explanations. Widely favored among Class 12 Science students, these solutions for Inverse Trigonometric Functions in Mathematics prove invaluable for efficiently completing homework assignments and preparing for exams. All queries and their corresponding answers from Chapter 2 of the NCERT Book for Class 12 Science Mathematics are generously provided on this platform, completely free of charge.

Page No 41:

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 

Page No 41:

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.

Page No 41:

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 

Page No 41:

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 

Page No 41:

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 

Page No 41:

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 

Page No 42:

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 

Page No 42:

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 

Page No 42:

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 

Page No 42:

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 

Page No 42:

Question 11:

Find the value of 

ANSWER:

Page No 42:

Question 12:

Find the value of 

ANSWER:

Page No 42:

Question 13:

Find the value of if sin⁻¹x = y, then

(A) (B) 

(C) (D) 

ANSWER:

It is given that sin⁻¹x = y.

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

Therefore,.

Page No 42:

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 = sinθ. Then, 

We have,

R.H.S. =

= 3θ

= L.H.S.

Page No 47:

Question 2:

Prove 

ANSWER:

To prove:

Let x = cosθ. Then, cos⁻¹x =θ.

We have,

Page No 47:

Question 3:

Prove 

ANSWER:

To prove:

Page No 47:

Question 4:

Prove 

ANSWER:

To prove: 

Page No 47:

Question 5:

Write the function in the simplest form:

ANSWER:

Page No 47:

Question 6:

Write the function in the simplest form:

ANSWER:

Put x = cosec θ ⇒ θ = cosec⁻¹x

Page No 47:

Question 7:

Write the function in the simplest form:

ANSWER:

Page No 47:

Question 8:

Write the function in the simplest form:

ANSWER:

Page No 48:

Question 9:

Write the function in the simplest form:

ANSWER:

Page No 48:

Question 10:

Write the function in the simplest form:

ANSWER:

Page No 48:

Question 11:

Find the value of 

ANSWER:

Let. Then,

Page No 48:

Question 12:

Find the value of 

ANSWER:

Page No 48:

Question 13:

Find the value of 

ANSWER:

Let x = tan θ. Then, θ = tan⁻¹x.

Let y = tan Φ. Then, Φ = tan⁻¹y.

Page No 48:

Question 14:

If, then find the value of x.

ANSWER:

On squaring both sides, we get:

Hence, the value of x is

Page No 48:

Question 15:

If, then find the value of x.

ANSWER:

Page No 48:

Question 16:

Find the values of 

ANSWER:

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

Here,

Now, can be written as:

Page No 48:

Question 17:

Find the values of 

ANSWER:

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

Here,

Now, can be written as:

Page No 48:

Question 18:

Find the values of 

ANSWER:

Let. Then,

Page No 48:

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 ⁻¹x.

Here,

Now, can be written as:

The correct answer is B.

Page No 48:

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.

Page No 48:

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 ⁻¹x.

Here,

Now, can be written as:

Page No 51:

Question 2:

Find the value of 

ANSWER:

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

Here,

Now, can be written as:

Page No 51:

Question 3:

Prove 

ANSWER:

Now, we have:

Page No 51:

Question 4:

Prove 

ANSWER:

Now, we have:

Page No 51:

Question 5:

Prove 

ANSWER:

Now, we will prove that:

Page No 51:

Question 6:

Prove 

ANSWER:

Now, we have:

Page No 51:

Question 7:

Prove 

ANSWER:

Using (1) and (2), we have

Page No 51:

Question 8:

Prove 

ANSWER:

Page No 52:

Question 9:

Prove 

ANSWER:

Page No 52:

Question 10:

Prove 

ANSWER:

Page No 52:

Question 11:

Prove  [Hint: putx = cos 2θ]

ANSWER:

Page No 52:

Question 12:

Prove 

ANSWER:

Page No 52:

Question 13:

Solve

ANSWER:

Page No 52:

Question 14:

Solve

ANSWER:

Page No 52:

Question 15:

Solveis equal to

(A)  (B)  (C)  (D) 

ANSWER:

Let tan⁻¹x = y. Then, 

The correct answer is D.

Page No 52:

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.

Page No 52:

Question 17:

Solveis equal to

(A) (B). (C) (D) 

ANSWER:

Hence, the correct answer is C.