Question:
Show that $\sin ^{-1}\left(2 x \sqrt{1-x^{2}}\right)=2 \sin ^{-1} x$.
Solution:
We have
LHS $=\sin ^{-1}\left(2 x \sqrt{1-x^{2}}\right)$ Putting $x=\sin a$, we get $=\sin ^{-1}\left(2 \sin a \sqrt{1-\sin ^{2} a}\right)$
$=\sin ^{-1}(2 \sin a \cos a)=\sin ^{-1}(\sin 2 a)=2 a=2 \sin ^{-1} x \quad(\because x=\sin a)$