Prove that:

To Prove: cos^{-1} (1 – 2x^{2}) = 2 sin^{-1} x

Formula Used: cos 2A = 1 – 2 sin^{2} A

Proof:

LHS = cos^{-1} (1 – 2x^{2}) … (1)

Let x = sin A … (2)

Substituting (2) in (1),

LHS = cos^{-1} (1 – 2 sin^{2} A)

= cos^{-1} (cos 2A)

= 2A

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

2A = 2 sin^{-1} x

= RHS

Therefore, LHS = RHS

Hence proved.

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