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

Lionel Ponton

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 trigon…