What is the simplest way to show that $\cos(r \pi)$ is irrational if $r$ is rational and $\displaystyle r \in \left(0,\frac{1}{2} \right)\setminus \left\{\frac{1}{3} \right\}$?
I proved it using the following sequence $x_1 = \cos(r \pi)$; $x_{k} = 2 x_{k-1}^2-1$ and periodicity of the cosine function. Is there any proof that is based on definition of rational numbers and trigonometric identities only? Thanks!