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.