Recurrent proofs of the irrationality of certain trigonometric values

TL;DR

This paper introduces elementary recurrence-based proofs for the irrationality of pi, tangent, and cosine of rational numbers, providing new insights into the nature of these fundamental constants.

Contribution

It presents novel, elementary recurrence methods to prove irrationality of key trigonometric values, expanding understanding beyond previous complex proofs.

Findings

Proves pi is irrational using recurrences of integrals.

Shows tan(r) is irrational for all nonzero rational r.

Establishes cos(r) is irrational for all nonzero rational r^2.

Abstract

We use recurrences of integrals to give new and elementary proofs of the irrationality of pi, tan(r) for all nonzero rational r, and cos(r) for all nonzero rational r^2. Immediate consequences to other values of the elementary transcendental functions are also discussed.