Question

Prove that:

sin ^-1 (3x – 4x³) = 3 sin ^-1 x, |x| ≤ 1/√2.

Answer 1

sin ^-1 (3x – 4x³) = 3 sin ^-1 x, |x| ≤ 1/√2

Take x = sin θ

Where θ = sin ^-1 x

Here LHS = sin ^-1 (3x – 4x³)

By substituting the value of x

= sin ^-1 (3 sin θ – 4 sin³θ )

So we get

= sin ^-1 (sin3θ)

= 3 θ

By substituting the value of θ

= 3 sin ^-1 x

= RHS

Hence, it is proved.