Class 12 NCERT Solutions- Mathematics Part I – Chapter 2 Inverse Trigonometric Functions – Miscellaneous Exercise on Chapter 2 | Set 1

Question 1. Find the value of 

Solution: 

We know that  

Here, 

Now,  can be written as :

, where 

Hence, the value of  = π/6

Question 2. Find the value of 

Solution: 

We know that 

Here, 

Now,  can be written as:

      

 where 

Hence, the value of  = π/6

Question 3. Prove 

Solution: 

Let           -(1)

sin x = 3/5

So,= 4/5

tan x = 3/4

Hence, 

Now put the value of x from eq(1), we get

Now, we have

L.H.S 

=                 –

Hence, proved.

Question 4. Prove 

Solution: 

Let  

Then sin x = 8/17 

 cos x = = 15/17

Therefore, 

         -(1)

Now, let 

Then, sin y = 3/5 

 = 4/5

     -(2)

Now, we have:

L.H.S.

 From equation(1) and (2), we get

=                          

= 

=                                  –

= 

Hence proved

Question 5. Prove 

Solution: 

Let 

Then, cos x = 4/5

 = 3/5

        -(1)

Now let 

Then, cos y = 3/4 

            -(2)

Let  

Then, cos z = 33/65

sin z = 56/65 

    -(3)

Now, we will prove that : 

L.H.S. 

From equation (1) and equation (2)

=                                                                  

=          –

= 

= 

Using equation(3)

=                                                                                                   

Hence proved

Question 6. Prove 

Solution: 

Let   

Then, sin x = 3/5 

 = 4/5

           -(1)

Now, let   

Then, cos y = 12/13 and sin y = 5/13

        -(2)                                                             

Let 

Then, sin z = 56/65 and cos z = 33/65

           -(3)

Now, we have:

L.H.S.=

From equation(1) and equation(2)

=                                                                           

=                –

= 

= 

From equation (3)

=     

Hence proved                                                                                                           

Question 7. Prove 

Solution:

Let    

Then, sin x = 5/13 and cos x = 12/13.

                   -(1)

Let      

Then, cos y = 3/5 and sin y = 4/5

                -(2)

From equation(1) and (2), we have

R.H.S.

=

=           –

=

=

L.H.S = R.H.S

Hence proved

Question 8. Prove 

Solution:

L.H.S.

=      –

= 

= 

= 

= 

= 

= 

= π/4 

L.H.S = R.H.S

Hence proved

Question 9. Prove 

Solution:

Let x = tan²θ

Then,

Now, we have

R.H.S = 

L.H.S = R.H.S

Hence proved

Question 10. Prove 

Solution:

Consider

By rationalizing

=                         

=

=

= 

L.H.S = = x/2

L.H.S = R.H.S

Hence proved

Chapter 2 Inverse Trigonometric Functions – Miscellaneous Exercise on Chapter 2 | Set 2