Answer
See below.
Work Step by Step
$LHS=\frac{sin(\alpha)cos(\beta)+cos(\alpha)sin(\beta)}{sin(\alpha)cos(\beta)-cos(\alpha)sin(\beta)}\times\frac{\frac{1}{cos(\alpha)cos(\beta)}}{\frac{1}{cos(\alpha)cos(\beta)}}=\frac{tan(\alpha)+tan(\beta)}{tan(\alpha)-tan(\beta)}=RHS$