Cosine and Sine of Sums

Question

What's a good way to simplify $\sin( \sum\nolimits_{i=1}^{\infty} x_i)$ as the product and sum of $\sin(x_i)$ and $\cos(x_i)$ alone? And the same for $\cos( \sum\nolimits_{i=1}^{\infty} x_i)$?

Answer 1

If the series $S = \sum_{j=1}^\infty x_j$ converges, then $$\cos(S) + i \sin(S) = \exp(i S) = \prod_{j=1}^\infty \exp(i x_j) = \prod_{j=1}^\infty (\cos(x_j) + i \sin(x_j))$$ the infinite product converging as well.

EDIT: In the finite case, $\cos\left(\sum_{j=1}^n x_j\right)$ is the sum of $(-1)^{|S|/2} \left(\prod_{j \notin S} \cos(x_j)\right) \left(\prod_{j \in S} \sin(x_j)\right)$ over all subsets $S$ of $\{1, \ldots, n\}$ with even cardinality, where $|S|$ is the cardinality of $S$. For example, in the case $n=4$ these subsets are $\emptyset$, six pairs, and $\{1,2,3,4\}$, so that $$\cos(x_1 + x_2 + x_3 + x_4) = \cos(x_1)\cos(x_1)\cos(x_3)\cos(x_4) - \cos(x_1)\cos(x_2)\sin(x_3)\sin(x_4) - \ldots - \cos(x_3)\cos(x_4)\sin(x_1)\sin(x_2) + \sin(x_1)\sin(x_2)\sin(x_3)\sin(x_4)$$

Similarly $\sin\left(\sum_{j=1}^n x_j\right)$ is the sum of $(-1)^{(|S|-1)/2} \left(\prod_{j \notin S} \cos(x_j)\right) \left(\prod_{j \in S} \sin(x_j)\right)$ over all subsets $S$ of odd cardinality.

Answer 2

I'll suppose that you are after $\cos \left( \sum_{k=0}^n x_k \right)$. Then, we have:

$$\cos \left( \sum_{k=0}^n x_k \right) = \Re \left( e^{i \sum_{k=0}^n x_k } \right) = \Re \left( \prod_{k=0}^n e^{i x_k} \right) $$

I don't see how this could be developed any further.

But I think that $\cos \left( \sum_{k=0}^n x_k \right)$ is generally easier to compute than any other formulas you might get (say by repeatedly applying $\cos(x+y) = \cos \cos - \sin \sin$).