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