Prove that:

To Prove: sin^-1 (3x – 4x³) = 3 sin^-1 x

Formula Used: sin 3A = 3 sin A – 4 sin³ A

Proof:

LHS = sin^-1 (3x – 4x³) … (1)

Let x = sin A … (2)

Substituting (2) in (1),

LHS = sin^-1 (3 sin A – 4 sin³ A)

= sin^-1 (sin 3A)

= 3A

From (2), A = sin^-1 x,

3A = 3 sin^-1 x

= RHS

Therefore, LHS = RHS

Hence proved.