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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 $\cos(r\pi)$ areirrational numbers except in usual cases.

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