Answer
$\cos^2 5x-\sin^2 5x=\cos 10x$
Work Step by Step
Start from the right side, write $10x$ as $2*5x$, and use the double-angle formula $\cos 2s=\cos^2 s-\sin^2 s$ with $s=5x$:
$\cos 10x$
=$\cos (2*5x)$
$=\cos^2 5x-\sin^2 5x$
Since this equals the left side, the identity is proven.