MP Board Class 12th Maths Important Questions Chapter 2 Inverse Trigonometric Functions

October 15, 2025

Important Questions

Very Short Answer Type Questions

Question 1.
Find the principle value of the following?

Question 2.
Prove the following:

Question 7.
Prove that: 3 sin^-1 x = sin^-1 (3x – 4x³)? (NCERT, CBSE 2018)
Solution:
Let sin^-1 x = θ
⇒ x = sin θ
We know that sin 3θ = 3 sinθ – 4 sin³ θ
= 3x – 4x³
⇒ 3θ = sin^-1 ( 3x – 4x³)
⇒ 3.sin^-1 x = sin^-1(3x – 4x³). proved.

Question 8.
Prove that: 3 cos^-1 x = cos^-1 (4x³ – 3x)? (NCERT)
Solution:
Let cos^-1 x = cos θ
⇒ x = cos θ
We know that cos 3θ = 4 cos³θ – 3 cos θ
= 4x³ – 3x
⇒ 3θ = cos^-1 (4x³ – 3x)
⇒ 3 cos^-1x = cos^-1 (4x³ – 3x). Proved.

Question 9.
Prove the following:

Question 10.
Prove that:

sec^-1 x + cosec^-1 x =
sin^-1x + cos^-1x =
tan^-1x + cot^-1x =

Solution:
1. sec^-1 x + cosec^-1x =
Let sec ^-1 x = θ
∴x = sec θ
⇒ x = cosec ( – θ)
⇒ cosec ^-1 x = – θ. Proved.

2. sin^-1 x + cos^-1 x =
Let sin^-1 x = θ ……………………. (1)
⇒ x = sin θ
⇒ x = cos ( – θ)
⇒ cos ^-1 x = – θ ………………. (2)
Adding eqns (1) and (2),
sin ^-1 x + cos ^-1 x = θ + – θ
⇒ sin ^-1 x + cos^-1 x = π2 Proved.

3. tan ^-1 x + cot^-1 x =
Let tan ^-1 x = θ
⇒ x = tan θ
⇒ x = cot ( – θ)
⇒ cot ^-1 x = – θ
Adding eqns. (1) and (2),
tan ^-1 x + cot ^-1 x = . Proved.

Question 11.
Prove the following:

2. Solve like Q.No. 11 (A).

3. Solve like Q.No. 11(A).

Question 12.
Prove that:

2. Solve like Q.No. 12 (A).

3. Solve like Q.No. 12 (A).

Question 13.
tan^-1 1 + tan^-1 2 + tan^-1 3 = π?
Solution:
L.H.S. = tan^-1 1 + (tan^-1 2 + tan^-1 3)

= tan^-1 (1) + π + tan^-1 (-1)
= tan^-1 (1) + π – tan^-1 (1), [∵tan^-1 (-x) = – tan^-1 x]
= π = R.H.S. Proved.

Question 16.
Prove that:
sin (cos^-1 x ) = cos (sin^-1 x)?
Solution:
L.H.S. = sin(cos^-1 x)
= sin [ – sin ^-1 x],
[∵ sin^-1 x + cos^-1x = , cos^-1x = – sin^-1 x]
= cos (sin^-1 x), [ ∵sin (90° – θ) = cos θ ]
= R.H.S. Proved.

Question 17.
(A) Prove that:

Question 18.
Solve the equation:

Question 19.
solve the equation:

Question 20.
(A) Prove the following:

Question 23.
If tan^-1 a + tan^-1 b + tan^-1 c = then prove that ab + bc + ca = 1?
Solution:
tan^-1 a + tan^-1 b + tan^-1 c = , given
⇒ tan^-1 a + tan^-1 b + tan ^-1 c = tan^-1 a + cot^-1 a, [∵tan^-1 a + cot ^-1 a = ]
⇒ tan^-1 b + tan^-1c = cot^-1 a

⇒ ab + ca = 1 – bc
⇒ ab + bc + ca = 1. Proved.

Long Answer Type Questions

Question 1.
(A) Prove that:

(B) Solve the following equation:
sin^-1 x + sin^-1 (1 – x) = sin^-1
Solution:
Let sin^-1 x = α ∴ x = sin α
Here α + sin^-1 ( 1 – sin α) = sin^-1
⇒ α + sin^-1 ( 1 – sin α) = sin^-1 cos α
⇒ α + sin^-1 ( 1 – sin α) = sin^-1. sin ( – α)
⇒ α + sin^-1 ( 1 – sin α) = – α
⇒ sin^-1 ( 1 – sin α) = – 2α
⇒ 1 – sin α = sin ( – 2α)
⇒ 1 – sin α = cos 2α
⇒ 1 – cos 2α = sin α
⇒ 2 sin²α = sin α
⇒ sin α = ∴ α =
or x =

Question 2.

Question 3.
Write in simplest form:

Question 4.
(A) Prove the following:

Question 5.
Solve the following equation:

Question 6.

Question 7.
If cos^-1 x + cos^-1 y + cos^-1 z = π then prove that:
x² + y² + z² + 2xyz = 1?
Solution:
Given: cos^-1 x + cos^-1 z = π
⇒ cos^-1 x + cos^-1 y = π- cos^-1 z

Squaring on both sides
x²y² + z² + 2xyz = (1 – x²) (1 – y²)
⇒ x²y² + z² + 2xyz = 1 – y² – x² + x²y²
⇒ z² + 2xyz = 1 – y² – x²
⇒ x² + y² + z² + 2xyz = 1. Proved.

Question 12.
Prove that:

= tan^-1 (tan 3θ) – tan^-1 (tan 2θ)
= 3θ – 2θ
= θ
= tan^-1 2x
= R.H.S. Proved.

Question 13.
Prove that:

Objective Type Questions:

Question 1.
Choose the correct answer:

Question 1.
If sin^-1x – cos ^-1 x = , then the value of x is equal to:

Question 2.
If tan^-13 + tan^-1 8, then the value of x is equal to:

Question 4.
The value of 2 tan^-1 {cosec (tan^-1 x ) – tan (cot^-1 x)} is equal to:
(a) cot^-1x
(b) cot^-11/x
(c) tan^-1x
(d) tan^-11/x
Answer:
(c) tan^-1x