Answer
$\sin 2x=2\sin x\cos x$
Work Step by Step
We are trying to use $\sin(x+y)=\sin x\cos y+\cos x\sin y$ to prove $\sin 2x=2\sin x\cos x$.
Start with the left side of what we're trying to prove:
$\sin 2x$
Write $2x$ as $x+x$ and use the Addition Formula:
$=\sin(x+x)$
$=\sin x\cos x+\cos x\sin x$
$=\sin x\cos x+\sin x\cos x$
$=2\sin x\cos x$
Since this equals the right side, the identity has been proven.