If `tan^(-1){(sqrt(1+x^2)-sqrt(1-x^2))/(sqrt(1+x^2)+sqrt(1-x^2))}=alpha,` then prove that `x^2=sin2alpha`

Explanation

To prove that \( x^2 = \sin(2\alpha) \) given that

\[ \tan^{-1}\left(\frac{\sqrt{1+x^2}-\sqrt{1-x^2}}{\sqrt{1+x^2}+\sqrt{1-x^2}}\right) = \alpha, \]

we will start by simplifying the left-hand side.

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