TITLE: Sine of a finite sum of angles
QUESTION [2 upvotes]: There is a formula for evaluating the function
$$
\sin\left(\sum_{i = 1}^\infty \theta_i\right),
$$
where $\theta_i$ form an absolutely convergent series. See Wikipedia. How to write this down for
$$
\sin\left(\sum_{i = 1}^N \theta_i\right),
$$
and how to prove it?
Any reference to an existing proof is highly appreciated.

REPLY [2 votes]: $$\eqalign{\sin\left(\sum_{j=1}^N \theta_j\right) &= \left(\exp\left(i \sum_{j=1}^N \theta_j \right) - \exp\left(-i \sum_{j=1}^N \theta_j\right)\right)/(2i)\cr
&= \left(\prod_{j=1}^N (\cos(\theta_j)+i\sin(\theta_j)) - \prod_{j=1}^N (\cos(\theta_j)-i\sin(\theta_j))\right)/(2i)\cr
&= \sum_{S \subseteq \{1,\ldots,N\}:\; |S|\ \text{odd}} (-1)^{(|S|-1)/2} \prod_{j \in S} \sin(\theta_j) \prod_{j \notin S} \cos(\theta_j)\cr}$$
the sum being over all subsets $S$ of $\{1,\ldots,N\}$ with odd cardinality.
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