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

#### Question 1:

Find the principal value of

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

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)

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

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

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)

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

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

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

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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

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

#### Question 12:

Find the value of

#### Question 13:

Find the value of if sin−1 y, then

(A)  (B)

(C)  (D)

It is given that sin−1 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)

#### Question 1:

Prove

To prove:

Let x = sinθ. Then,

We have,

R.H.S. =

= 3θ

= L.H.S.

#### Question 2:

Prove

To prove:

Let x = cosθ. Then, cos−1 x =θ.

We have,

Prove

To prove:

Prove

To prove:

#### Question 5:

Write the function in the simplest form:

#### Question 6:

Write the function in the simplest form:

Put x = cosec θ ⇒ θ = cosec−1 x

#### Question 7:

Write the function in the simplest form:

#### Question 8:

Write the function in the simplest form:

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

#### Question 9:

Write the function in the simplest form:

#### Question 10:

Write the function in the simplest form:

#### Question 11:

Find the value of

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Let. Then,

#### Question 12:

Find the value of

#### Question 13:

Find the value of

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.

On squaring both sides, we get:

Hence, the value of x is

#### Question 15:

If, then find the value of x.

Hence, the value of x is

#### Question 16:

Find the values of

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

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

Let. Then,

#### Question 19:

Find the values of is equal to

(A)  (B)  (C)  (D)

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

#### Question 20:

Find the values of is equal to

(A)  (B)  (C)  (D) 1

Let. Then,

We know that the range of the principal value branch of.

#### Question 21:

Find the values of is equal to

(A) π (B)  (C) 0 (D)

Let. Then,

We know that the range of the principal value branch of

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Let.

The range of the principal value branch of

#### Question 1:

Find the value of

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

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

Here,

Now,

can be written as:

Prove

Now, we have:

Prove

Now, we have:

#### Question 5:

Prove

Now, we will prove that:

Prove

Now, we have:

#### Question 7:

Prove

Using (1) and (2), we have

Prove

Prove

Prove

#### Question 11:

Prove  [Hint: putx = cos 2θ]

Prove

Solve

Solve

#### Question 15:

Solveis equal to

(A)  (B)  (C)  (D)

Let tan−1 x = y. Then,

#### Question 16:

Solvethen x is equal to

(A)  (B)  (C) 0 (D)

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)