​ Solve: 3sin^-1x = sin^-1(3x - 4x^3), x ∈ [-1/2, 1/2] ​

Question

Solve: 3sin^-1x = sin^-1(3x - 4x³), x ∈ [-\(\frac{1}{2}\), \(\frac{1}{2}\)]

Answer 1

Let sin^-1 x be θ 

⇒ x = sin θ 

In RHS putting the value of x 

sin^-1(3x – 4x³) 

⇒ sin^-1(3sinθ – 4sin³θ) 

= sin^-1(sin3θ) 

[∵ 3sinθ – 4sin³θ = sin3θ] = 3θ 

putting the value of θ 

3θ = 3sin^-1 x = LHS 

∴ 3sin^-1 x = sin^-1 (3x – 4x³) 

Hence proved.