Answer
$\frac{\sin 10x}{\sin 9x+\sin x}=\frac{\cos 5x}{\cos 4x}$
Work Step by Step
Start with the left side:
$\frac{\sin 10x}{\sin 9x+\sin x}$
Use the identity $\sin x+\sin y=2\sin\frac{x+y}{2}\cos \frac{x-y}{2}$ to expand:
$=\frac{\sin 10x}{2\sin\frac{9x+x}{2}\cos \frac{9x-x}{2}}$
$=\frac{\sin 10x}{2\sin 5x\cos 4x}$
Use the identity $\sin 2A=2\sin A\cos A$ to expand:
$=\frac{\sin (2*5x)}{2\sin 5x\cos 4x}$
$=\frac{2\sin 5x\cos 5x}{2\sin 5x\cos 4x}$
$=\frac{\cos 5x}{\cos 4x}$
Since this equals the right side, the identity has been proven.
