A direct proof of the irrationality of $\tan^2(r \pi)$

Abstract

Given a rational number $r$ such that $2r$ is not an integer, we prove that $\tan^2(r\pi)$ is irrational unless it is equal to $0$, $1$, $3$ or $\frac{1}{3}$, using only basic trigonometry and the Rational Root Theorem. Moreover, we deduce that $\tan(r\pi)$, $\cos^2(r\pi)$ ans $\cos(r\pi)$ are irrational numbers except in usual cases.