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Prove that sin -1 (2x √1 – x2) = 2 sin -1 x, |x| ≤ 1/√2.

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Best answer

Take x = sin θ

Where θ = sin -1 x

Here LHS = sin -1 (2x √1 – x2)

So we get

= sin -1 (2 sin θ √1 – cos2 θ)

It can be written as

= sin -1 (2 sin θ cos θ)

We know that sin 2θ = 2 sin θ cos θ

= sin -1 (sin 2θ)

= 2 θ

Substituting the value of θ

= 2 sin -1 x

= RHS

Hence, it is proved.

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