I read somewhere that $\cos(\theta \pi)$ is inside the splitting field of $x^n - 1$. (Assuming that $\theta$ is rational.) I know this field is $\mathbb{Q}(\zeta_n)$ where $\zeta_n$ is a primitive $n$-th root of unity. This means that we should be able to write $\cos(\theta \pi) = c_0 + c_1\zeta_n + \dots + c_{n - 1}\zeta^{n - 1}$ where $c_i \in \mathbb{Q}$. However, I don't know how I can find these coefficients.
My attempt:
First assume that $\theta = \frac{m}{n}$. We have $\zeta_n = e^{\frac{2\pi i}{n}}$. We know $\cos(\pi \theta) + i\sin(\pi \theta) = e^{\frac{m}{n}\pi i}$. If I manage to write $\cos(\pi\theta) - i\sin(\pi \theta)$ as a power of $e$, I might be able to add these two and write $2\cos(\pi \theta)$ as an element of $\mathbb{Q}(\zeta_n)$ using the elements $\{1, \zeta_n, \dots, \zeta_{n}^{n - 1}\}$ but I am stuck here.
Hints would be appreciated and thanks in advance.
Note: I found this question. H.Linkhorn has stated in the problem that they can write $\cos(\frac{2}{5}\pi)$ using the $5$-th roots of unity but I didn't understand the process.