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

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 

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 = sinθ. Then, 

We have,

R.H.S. =

= 3θ

= L.H.S.

Question 2:

Prove 

Answer:

To prove:

Let x = cosθ. 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⁻¹x.

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⁻¹x.

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 ⁻¹x.

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 ⁻¹x.

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 ⁻¹x.

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: putx = cos 2θ]

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.