Question

Prove the following:

3 sin⁻¹ x = sin⁻¹ (3x − 4x³), `x ∈ [-1/2, 1/2]`

Answer 1

Let x = sin θ.

Then, sin⁻¹ x = θ.

We have,

R.H.S = sin⁻¹ (3x – 4x³) = sin⁻¹ (3 sin θ – 4 sin³θ)

= sin⁻¹ (sin 3θ) = sin⁻¹ (3 sin θ – 4 sin³θ)

= 3θ = sin⁻¹ (3 sin θ – 4 sin³θ)

= 3 sin⁻¹ x = sin⁻¹ (3 sin θ – 4 sin³θ)

R.H.S = L.H.S