Expansion for $\cos(nx)$, with $n \in \mathbb{R}$

Question

Is there an expansion for $\cos(nx)$ when $n$ is not an integer? I'm trying to rewrite the following equation

Equation = $cos(nx_{1}) + cos(nx_{2}) + cos(nx_{3}) + ... + cos(nx_{600})$

as

Equation = factor $[cos(x_{1}) + cos(x_{2}) + cos(x_{3}) + ... + cos(x_{600})]$

Answer 1

I believe there is no such expansion. Indeed, any polynomial expression in $\cos(x)$ and $\sin(x)$ will be $2\pi$-periodic. But if $\alpha \in \mathbb{R}$ is not an integer, then $x \mapsto \cos(\alpha x)$ is not $2\pi$-periodic, hence it cannot be rewritten as a polynomial in $\cos(x)$ and $\sin(x)$.

Now, if you allow roots of polynomials then, by writing $\cos(x) = T_n(\cos(x/n))$ (with $T_n$ the Chebychev polynomial), we can express $\cos(x/n)$ as a root of $T_n - \cos(x)$ (which manageable for $n=2$ but gets complicated fast).