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#### 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 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^{−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,.

#### 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* = sin*θ*.
Then,

We have,

R.H.S. =

=
3*θ*

= L.H.S.

#### Page No 47:

#### Question 2:

Prove

#### Answer:

To prove:

Let *x* = cos*θ*.
Then, cos^{−1} *x* =*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^{−1}
*x*

#### 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:

${\mathrm{tan}}^{-1}\left(\frac{\mathrm{cos}x-\mathrm{sin}x}{\mathrm{cos}x+\mathrm{sin}x}\right)\phantom{\rule{0ex}{0ex}}={\mathrm{tan}}^{-1}\left(\frac{1-{\displaystyle \frac{\mathrm{sin}x}{\mathrm{cos}x}}}{1+{\displaystyle \frac{\mathrm{sin}x}{\mathrm{cos}x}}}\right)\phantom{\rule{0ex}{0ex}}={\mathrm{tan}}^{-1}\left(\frac{1-\mathrm{tan}x}{1+\mathrm{tan}x}\right)\phantom{\rule{0ex}{0ex}}={\mathrm{tan}}^{-1}\left(1\right)-{\mathrm{tan}}^{-1}\left(\mathrm{tan}x\right)\left({\mathrm{tan}}^{-1}\frac{x-y}{1+xy}={\mathrm{tan}}^{-1}x-{\mathrm{tan}}^{-1}y\right)\phantom{\rule{0ex}{0ex}}=\frac{\mathrm{\pi}}{4}-x$

#### 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^{−1}
*x*.

Let *y* = tan *Φ*.
Then, *Φ* = tan^{−1}
*y*.

#### 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:

Hence, the value of *x*
is

#### 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^{−1}*x*.

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^{−1}*x*.

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 ^{−1}*x*.

Here,

Now, can be written as:

${\mathrm{cos}}^{-1}\left(\mathrm{cos}\frac{7\mathrm{\pi}}{6}\right)={\mathrm{cos}}^{-1}\left[\mathrm{cos}\left(\mathrm{\pi}+\frac{\mathrm{\pi}}{6}\right)\right]\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}{\mathrm{cos}}^{-1}\left(\mathrm{cos}\frac{7\mathrm{\pi}}{6}\right)={\mathrm{cos}}^{-1}\left[-\mathrm{cos}\frac{\mathrm{\pi}}{6}\right]\left[\mathrm{as},\mathrm{cos}\left(\mathrm{\pi}+\mathrm{\theta}\right)=-\mathrm{cos}\mathrm{\theta}\right]\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}{\mathrm{cos}}^{-1}\left(\mathrm{cos}\frac{7\mathrm{\pi}}{6}\right)={\mathrm{cos}}^{-1}\left[-\mathrm{cos}\left(\mathrm{\pi}-\frac{5\mathrm{\pi}}{6}\right)\right]\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}{\mathrm{cos}}^{-1}\left(\mathrm{cos}\frac{7\mathrm{\pi}}{6}\right)={\mathrm{cos}}^{-1}\left[-\left\{-\mathrm{cos}\left(\frac{5\mathrm{\pi}}{6}\right)\right\}\right]\left[\mathrm{as},\mathrm{cos}\left(\mathrm{\pi}-\mathrm{\theta}\right)=-\mathrm{cos}\mathrm{\theta}\right]\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}$

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 ^{−1}*x*.

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 ^{−1}*x*.

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: **put*x* =
cos 2*θ*]

#### 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^{−1}
*x* = *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**.

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