Answer
See proof
Work Step by Step
$\cos2x=1-2\sin^2x$
$\bf{Solution:}$
\begin{align*}
\text{We know that}\\
\cos2x=&\cos(x+x)\\
=& \cos x\cos x-\sin x\sin x ~~\text{Cosine Sum Identity}\\
=&\cos^2x-\sin x\sin x ~~ \because \cos x\cos x=\cos^2x\\
=&\cos^2x-\sin^2 x ~~~~~~~~~ \because\sin x \sin x =\sin^2x\\
=&(1-\sin^2x)-\sin^2x ~~\because \cos^2 x = 1 - \sin^2 x\\
=&1-2\sin^2x ~~~~\text{ Simplify}
\end{align*}
$\text{The left side is identical to the right side, so the given equation is an identity.}$
