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