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.

arXiv comment2 pages