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

Put `x^{2}=\sin (2 \theta)`

Now, `\tan ^{-1}\left(\frac{\sqrt{1+\sin 2 \theta}-\sqrt{1-\sin 2 \theta}}{\sqrt{1+\sin 2 \theta}+\sqrt{1-\sin 2 \theta}}\right)=\alpha`

`\Rightarrow \tan ^{-1}\left(\frac{(\cos \theta+\sin \theta)-(\cos \theta-\sin \theta)}{(\cos \theta+\sin \theta)+(\cos \theta-\sin \theta)}\right)=\alpha`

`\Rightarrow \tan ^{-1}\left(\frac{\sin \theta}{\cos \theta}\right)=\alpha`

`\Rightarrow \quad \tan ^{-1}(\tan \theta)=\alpha`

`\Rightarrow \quad \theta=\alpha`

`\Rightarrow \quad 2 \theta=2 \alpha`

`\Rightarrow \quad \sin (2 \theta)=\sin (2 \alpha)`

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