Prove that

$=\frac{\sin x+\sin y}{\sin x-\sin y}$

$=\frac{2 \sin \frac{x+y}{2} \cos \frac{x-y}{2}}{2 \cos \frac{x+y}{2} \sin \frac{x-y}{2}}$

$=\tan \frac{x+y}{2} \cot \frac{x-y}{2}$

Using the formula,

$\sin A+\sin B=2 \sin \frac{A+B}{2} \cos \frac{A-B}{2}$

$\sin A-\sin B=2 \cos \frac{A+B}{2} \sin \frac{A-B}{2}$