Answer
$\sin(x+y)-\sin(x-y)=2\cos x\sin y$
Work Step by Step
Start with the left side:
$\sin(x+y)-\sin(x-y)$
Expand using the addition and subtraction formulas for sine:
$=(\sin x\cos y+\cos x\sin y)-(\sin x\cos y-\cos x\sin y)$
Simplify:
$=\sin x\cos y+\cos x\sin y-\sin x\cos y+\cos x\sin y$
$=\cos x\sin y+\cos x\sin y$
$=2\cos x\sin y$
Since this equals the right side, the identity has been proven.
