# NCERT Solutions for Miscellaneous Exercise Chapter 2 Class 12 - Inverse Trigonometric Functions

NCERT solutions for Class 12 Maths chapter 2 miscellaneous exercise has a set of questions which are not introduced in earlier exercises. Class 12 Maths chapter 2 miscellaneous exercise basically deals with the questions related to proving of the inverse trigonometric functions equivalent. By using various methods and techniques, such equations are proved equal to each other. Students can easily find that many questions were asked from Class 12 Maths chapter 2 miscellaneous exercise in the board exams. Hence it is highly recommended to practice the NCERT solutions for Class 12 maths chapter 2 including miscellaneous exercise which is present in NCERT Class 12th book. Students can also refer to NCERT questions for better understanding.

NCERT solutions for class 12 maths chapter 2 exercise 2.1

NCERT solutions for class 12 maths chapter 2 exercise 2.2

**NCERT Solutions For Class 12 Maths Chapter 2 Inverse Trigonometric Functions: Miscellaneous Exercise**

** ** **Question:1 ** Find the value of the following:

** Answer: **

If then , which is principal value of .

So, we have

** Therefore we have, **

** . **

** Question:2 ** Find the value of the following:

** Answer: **

We have given ;

so, as we know

So, here we have .

Therefore we can write as:

** . **

** Question:3 ** Prove that ** **

** Answer: **

To prove: ;

Assume that

then we have .

or

** Therefore we have **

** Now, **

** We can write L.H.S as **

** **

as we know

** L.H.S = R.H.S **

** Question:4 ** Prove that

** Answer **

Taking

then,

Therefore we have-

** .............(1). **

,

Then,

** .............(2). **

** So, we have now, **

** L.H.S. **

using equations (1) and (2) we get,

** = R.H.S. **

** Question:5 ** Prove that

** Answer: **

Take and and

then we have,

Then we can write it as:

or

** ...............(1) **

Now,

So, ** ...................(2) **

Also we have similarly;

Then,

** ...........................(3) **

Now, we have

** L.H.S **

so, using ** (1) ** and ** (2) ** we get,

or we can write it as;

** = R.H.S. **

** Hence proved. **

** Question:6 ** Prove that

** Answer: **

Converting all terms in ** tan ** form;

Let ** , ** and .

now, converting all the terms:

** or **

** We can write it in tan form as: **

** . **

** or ................(1) **

** or **

** We can write it in tan form as: **

** **

** or ......................(2) **

Similarly, for ** ; **

we have ** .............(3) **

** Using (1) and (2) we have L.H.S **

** On applying **

** We have, **

** ...........[Using (3)] **

** =R.H.S. **

** Hence proved. **

** Question:7 ** Prove that

** Answer: **

** Taking R.H.S; **

We have

** Converting sin and cos terms in tan forms: **

Let and

now, we have or

** ............(1) **

Now,

** ................(2) **

** Now, Using (1) and (2) we get, **

R.H.S.

as we know

so,

** equal to L.H.S **

** Hence proved. **

** Question:8 ** Prove that ** **

** Answer: **

Applying the formlua:

on two parts.

we will have,

** Hence it s equal to R.H.S **

** Proved. **

** Question:9 ** Prove that ** **

** Answer: **

By observing the square root we will first put

.

Then,

we have

or, ** R.H.S. **

.

** L.H.S. **

hence ** L.H.S. = R.H.S proved. **

** Question:10 ** Prove that

** Answer: **

Given that

By observing we can rationalize the fraction

We get then,

Therefore we can write it as;

** As L.H.S. = R.H.S. **

** Hence proved. **

** Question:11 ** Prove that ** **

[Hint: Put ]

** Answer: **

By using the Hint we will put ;

we get then,

dividing numerator and denominator by ,

we get,

using the formula

** **

** As L.H.S = R.H.S **

** Hence proved **

** Question:12 ** Prove that

** Answer: **

We have to solve the given equation:

** Take as common in L.H.S, **

or from

Now, assume,

Then,

Therefore we have now,

So we have L.H.S then

** That is equal to R.H.S. **

** Hence proved. **

** Question:13 ** Solve the following equations: ** **

** Answer: **

Given equation ;

Using the formula:

We can write

So, we can equate;

that implies that .

or or

** Hence we have solution . **

** Question:14 ** Solve the following equations:

** Answer: **

Given equation is

:

L.H.S can be written as;

Using the formula

So, we have

** Hence the value of . **

** Question:15 ** is equal to

(A)

(B)

(C)

(D)

** Answer: **

Let then we have;

or

** Hence the correct answer is D. **

** Question:16 ** then is equal to

(A)

(B)

(C) 0

(D)

** Answer: **

Given the equation:

we can migrate the term to the ** R.H.S. **

** then we have; **

or ** ............................(1) **

from

Take or .

So, we conclude that;

Therefore we can put the value of in equation ** (1) we get, **

Putting ** x= sin y ** , in the above equation; we have then,

So, we have the solution;

Therefore we have .

When we have , we can see that :

** So, it is not equal to the R.H.S. **

** Thus we have only one solution which is x = 0 **

** Hence the correct answer is (C). **

** Question:17 ** is equal to

(A)

(B)

(C)

(D)

** Answer: **

Applying formula: .

We get,

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

## More About NCERT Solutions for Class 12 Maths Chapter 2 Miscellaneous Exercise

The NCERT Class 12 Maths chapter Inverse Trigonometric functions provided here is prepared by the experienced faculties. NCERT solutions for Class 12 Maths chapter 2 miscellaneous exercise covers the major syllabus of this chapter from exam perspective. As questions from this exercise are asked more than those of previous exercises. Therefore NCERT solutions for Class 12 Maths chapter 2 miscellaneous exercise becomes a must to do exercise for the examination.

Also Read| Inverse Trigonometric Functions NCERT Notes

## Benefits of NCERT Solutions for Class 12 Maths Chapter 2 Miscellaneous Exercises

Miscellaneous exercise chapter 2 Class 12 NCERT syllabus has some of the questions which are very important from exam point of view.

Questions mentioned in NCERT book Class 12 Maths chapter 2 miscellaneous solutions are of the level of JEE and NEET.

It is a good source for revision also. Hence NCERT solutions for Class 12 Maths chapter 2 miscellaneous exercises can be referred directly to score well in the exam.

### Also see-

NCERT exemplar solutions Class 12 Maths chapter 2

NCERT solutions for Class 12 Maths chapter 2

NCERT Solutions Subject Wise

NCERT solutions class 12 Chemistry

NCERT solutions for class 12 Physics

NCERT solutions for class 12 Biology

NCERT solutions for class 12 Mathematics

## Subject wise NCERT Exemplar solutions

NCERT Exemplar Class 12th Maths

NCERT Exemplar Class 12th Physics

NCERT Exemplar Class 12th Chemistry

NCERT Exemplar Class 12th Biology

Happy learning!!!