Mathematics Trigonometry Multiple and Sub-multiple Angles

If sin α + sin β = p and cos α + cos β = q, then find the value of \(\rm {1\over 2}\ sin(\alpha + \beta) \).

1
\(\rm \frac{pq}{p^{2} + q^{2}}\)
2
\(\rm \frac{2pq}{p^{2} - q^{2}}\)
3
\(\rm \frac{-2pq}{p^{2} + q^{2}}\)
4
None of these

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