Prove that

$=\frac{\cos x-\cos y}{\cos y-\cos x}$

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

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

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

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

Using the formula,

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

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