I have a question and I would appreciate if you could give me some clues. For a given $\cos\theta_1,\cos\theta_2,\ldots,\cos\theta_n$, how could I calculate $\cos\left(\theta_1+\theta_2+\cdots+\theta_n\right)$? Are there some general equations or identities available?
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You can use the formula $\cos(a+b)=\cos(a)\cos(b)-\sin(a)\sin(b)$ several times to reduce to a polynomial in terms of $\sin(\theta_i)$ and $\cos(\theta_i)$ and finally use the equation $\sin^2(a)+\cos^2(b)=1$ to reduce everything in terms of $\cos(\theta_i)$'s.