If $\sin \alpha + \sin \beta = a, \; \cos \alpha + \cos \beta = b$, then $\sin (\alpha + \beta) \cos^2 (\frac{\alpha - \beta}{2})$ is equal to :

Question

( A ) $\frac{a+b}{2}$
( B ) $\frac{ab}{2}$
( C ) $\frac{2b}{a^2+b^2-1}$
( D ) $\frac{2ab}{a^2 - b^2}$