Answer
See below.
Work Step by Step
Using trigonometric identities, we have:
$LHS=\frac{sin(\alpha+\beta)}{tan\alpha+tan\beta}=\frac{sin\alpha cos\beta + cos\alpha sin\beta}{sin\alpha/cos\alpha+sin\beta/cos\beta}=\frac{sin\alpha cos\beta + cos\alpha sin\beta}{\frac{sin\alpha cos\beta + cos\alpha sin\beta}{cos\alpha cos\beta}}=cos\alpha cos\beta=RHS$
