Question

$\sin \left(2{\tan }^{-1}\sqrt{\frac{1-x}{1+x}}\right)$.

Answer 1

Putting $x=\cos 2\theta$

$\sin (2{\tan }^{-1}\sqrt{\frac{1-\cos 2\theta }{1+\cos 2\theta }})$

$\sin \left(2{\tan }^{-1}\tan \theta \right)$

$\sin 2\theta$

$\sqrt{1-{\cos }^{2}2\theta }$

$\sqrt{1-x^2}$