General formulas for $\cos \left(\sum_{i=1}^{n}\theta_{i}\right)$ and $\sin \left(\sum_{i=1}^{n}\theta_{i}\right)$?

Question

Can someone tell me the general formula for the cosine and sine of a sum of more than two angles? $$\cos \left(\sum_{i=1}^{n}\theta_{i}\right)=? \qquad \sin \left(\sum_{i=1}^{n}\theta_{i}\right)=?$$

I can't find it anywhere!

Answer 1

According to Wikipedia, $$ \cos\Bigl(\sum_{i=1}^{n}\theta_{i}\Bigr)=\sum_{\substack{0\le k\le n\\k\text{ even}}}(-1)^{k/2}\sum_{\substack{A\subset\{1,\dots,n\}\\|A|=k}}\Bigl(\prod_{i\in A}\sin\theta_i\prod_{i\notin A}\cos\theta_i\Bigr) $$