Prove that:

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

Formula Used: sin 3A = 3 sin A – 4 sin^{3} A

Proof:

LHS = sin^{-1} (3x – 4x^{3}) … (1)

Let x = sin A … (2)

Substituting (2) in (1),

LHS = sin^{-1} (3 sin A – 4 sin^{3} A)

= sin^{-1} (sin 3A)

= 3A

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

3A = 3 sin^{-1} x

= RHS

Therefore, LHS = RHS

Hence proved.

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