Prove 3 sin^{- 1} x = sin^{- 1} (3x - 4x³), x ∈ [- 1/2, 1/2]

Solution:

Inverse trigonometric functions are the inverse ratio of the basic trigonometric ratios. 

Here the basic trigonometric function of Sin θ = y can be changed to θ = sin^-1 y

Let x = sin θ

Hence, sin^{- 1} (x) = θ

Now,

RHS = sin^{- 1} (3x - 4x³)

Using trigonometric identity 

sin 3 θ = 3 sin θ - 4 sin³ θ

= sin^{- 1} (3sin θ - 4sin³ θ)

= sin^{- 1} (sin 3θ)

= 3 θ

= 3 sin^{- 1} x

= LHS

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NCERT Solutions for Class 12 Maths - Chapter 2 Exercise 2.2 Question 1

Prove 3 sin^{- 1} x = sin^{- 1} (3x - 4x³), x ∈ [- 1/2, 1/2]

Summary:

Here we have proved 3 sin^{- 1} x = sin^{- 1} (3x - 4x³).  Inverse trigonometric functions are the inverse ratio of the basic trigonometric ratios