Answer
See below.
Work Step by Step
Use the Sum-to-Product Formula, we have:
$LHS=\frac{cos(\alpha)+cos(\beta)}{cos(\alpha)-cos(\beta)}
=\frac{2cos(\frac{\alpha+\beta}{2})cos(\frac{\alpha-\beta}{2})}{-2sin(\frac{\alpha+\beta}{2})sin(\frac{\alpha-\beta}{2})}
=-cot(\frac{\alpha+\beta}{2})cot(\frac{\alpha-\beta}{2})=RHS$
