How to prove that $\cos(n)$ is irrational?

Question

We know that $\cos(1)$ is real and transcendental (1). Then by using the fact that for every $n \in \mathbb{N}$ there exists a polynomial $P_n$ of degree $n$ with integer coefficients such that $\cos(nt) =P_n(cos(t))$ for all $t \in \mathbb{R}$ , one can prove that $\cos(n)$ is irrational for all $n$. Is there an 'easy' way to prove this without relying on (1).

Answer 1

You can mimic the proof of the irrationality of $e$ relying on the Taylor series of the cosine function, for which: $$\cos(n)=\sum_{k=0}^{+\infty}\frac{(-1)^k n^{2k}}{(2k)!}.$$