Answer
$\frac{\sin x+\sin 5x}{\cos x+\cos 5x}=\tan 3x$
Work Step by Step
Start with the left side:
$\frac{\sin x+\sin 5x}{\cos x+\cos 5x}$
Use the identities $\sin x+\sin y=2\sin\frac{x+y}{2}\cos\frac{x-y}{2}$ and $\cos x+\cos y=2\cos\frac{x+y}{2}\cos\frac{x-y}{2}$ to expand:
$=\frac{2\sin\frac{x+5x}{2}\cos\frac{x-5x}{2}}{2\cos\frac{x+5x}{2}\cos\frac{x-5x}{2}}$
$=\frac{2\sin 3x\cos (-2x)}{2\cos 3x\cos(-2x)}$
$=\frac{\sin 3x}{\cos 3x}$
$=\tan 3x$
Since this equals the right side, the identity has been proven.
