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

TL;DR

This paper provides a straightforward proof that for rational multiples of pi, the square of the tangent is irrational unless it equals specific rational values, using basic trigonometry and the Rational Root Theorem.

Contribution

It offers a direct, elementary proof of the irrationality of $ an^2(r \pi)$ for rational $r$, extending to related trigonometric functions.

Findings

$ an^2(r\pi)$ is irrational unless it equals 0, 1, 3, or 1/3

$ an(r\pi)$, $ ext{cos}^2(r\pi)$, and $ ext{cos}(r\pi)$ are irrational in general

The proof uses only basic trigonometry and the Rational Root Theorem.

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.