Prove that:

To Prove: $\sin ^{-1}\left(2 x \sqrt{1-x^{2}}\right)=2 \sin ^{-1} x$

Formula Used: $\sin 2 A=2 \times \sin A \times \cos A$

Proof:

$\mathrm{LHS}=\sin ^{-1}\left(2 \mathrm{x} \sqrt{1-\mathrm{x}^{2}}\right) \cdots$(1)

Let $x=\sin A \ldots$ (2)

Substituting $(2)$ in $(1)$,

LHS $=\sin ^{-1}\left(2 \sin A \sqrt{1-\sin ^{2} A}\right)$

$=\sin ^{-1}(2 \times \sin A \times \cos A)$

$=\sin ^{-1}(\sin 2 A)$

$=2 A$

From $(2), A=\sin ^{-1} x$

$2 A=2 \sin ^{-1} x$

$=$ RHS

Therefore, LHS $=$ RHS

Hence proved.