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Class XII Chapter 2 – Inverse Trigonometric Functions Maths

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**Question 1:**

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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

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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 .

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

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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

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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

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

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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

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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

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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

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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

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Find the value of Answer

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Find the value of Answer

Find the value of if sin^{−1} *x *=
*y*, then

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**(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, .

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Find the value of is equal to

**(A)
**π (**B) **(**C) **(**D)**** **Answer

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**Question 1:**

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Prove Answer

To prove:

Let *x *= sin*θ*. Then, We have,

R.H.S. =

= 3*θ*

= L.H.S.

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Prove Answer

To prove:

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

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Prove Answer

To prove:

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Prove Answer

To prove:

Write the function in the simplest form:

Answer

Write the function in the simplest form:

Answer

Put *x *=
cosec *θ *⇒ *θ *= cosec^{−1} *x*

* *

Write the function in the simplest form:

Answer

Write the function in the simplest form:

Answer

Write the function in the simplest form:

Answer

Write the function in the simplest form:

Answer

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Find the value of Answer

Let . Then,

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Find the value of Answer

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Find the value of Answer

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

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

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If , then find the value of

On squaring both sides, we get:

Hence, the value of *x *is

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If , then find the value of *x*.

Answer

Hence, the value of *x *is

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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:

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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:

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Find the values of Answer

Let . Then,

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Find the values of is equal to

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.

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Find the values of is equal to

Answer

Let . Then,

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

∴

The correct answer is D.

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**Question 1:**

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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:

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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:

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Prove Answer

Now, we have:

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Prove Answer

Now, we have:

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Prove Answer

Now, we will prove that:

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Prove Answer

Now, we have:

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Prove Answer

Using (1) and (2), we have

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Prove Answer

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Prove Answer

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Prove Answer

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Prove [**Hint: **put*x *= cos 2*θ*]

Answer

Prove Answer

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Solve Answer

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Solve Answer

Solve is equal to

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

Let tan^{−1} *x *= *y*. Then,

The correct answer is D.

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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**.

Solve is equal to

Answer

Hence, the correct answer is **C**.