Prove that:

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

Formula Used: $\cos 2 A=1-2 \sin ^{2} A$

Proof:

$\mathrm{LHS}=\cos ^{-1}\left(1-2 \mathrm{x}^{2}\right) \ldots(1)$

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

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

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

$=\cos ^{-1}(\cos 2 \mathrm{~A})$

$=2 \mathrm{~A}$

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

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

$=\mathrm{RHS}$

Therefore, $\mathrm{LHS}=\mathrm{RHS}$

Hence proved.