  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.
