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