Gold ratio and a trigonometric identity

TL;DR

This paper presents two proofs of a trigonometric identity involving the golden ratio, one utilizing the golden ratio and one independent of it, to deepen understanding of their relationship.

Contribution

It provides novel proofs of a specific trigonometric identity, highlighting the connection between the golden ratio and cosine functions.

Findings

The identity holds true and can be proved with or without referencing the golden ratio.

Both proofs confirm the mathematical relationship involving cosine values at specific angles.

The work enhances understanding of the interplay between the golden ratio and trigonometric identities.

Abstract

We give two proofs of the identity $$\sqrt{\frac{\cos\frac{2\pi}{5}} {\cos\frac{\pi}{5}}}+\sqrt{\frac{\cos\frac{\pi}{5}} {\cos\frac{2\pi}{5}} }=\sqrt{5},$$ using and not using the gold ratio.