Answer
See the explanation below.
Work Step by Step
${{\sin }^{2}}x+\cos 2x={{\cos }^{2}}x$
Consider the left side of the given expression and apply the double angle formula.
$\begin{align}
& {{\sin }^{2}}x+\cos 2x={{\sin }^{2}}x+{{\cos }^{2}}x-{{\sin }^{2}}x \\
& ={{\cos }^{2}}x
\end{align}$
Hence, it is proved that the given identity ${{\sin }^{2}}x+\cos 2x={{\cos }^{2}}x$ holds true.