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

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).

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You can always just prove (1) by using Lindemann's theorem that if $\alpha$ is algebraic, $e^\alpha$ is transcendental. Then using that $i$ is algebraic and $e^{i}=\cos 1+i\sin 1$, you get (1). – Adam Hughes Jul 5 '14 at 19:17

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