# On certain q-trigonometric identities

## TL;DR

This paper introduces $q$-analogues of tangent and cotangent functions, establishing theta function identities and deriving new $q$-trigonometric identities related to classical identities.

## Contribution

It defines $q$-analogues for tangent and cotangent and uses elliptic function theory to derive new theta function identities and $q$-trigonometric identities.

## Key findings

- Derived two $q$-trigonometric identities involving $ an_q z$ and $ ext{cot}_q z$
- Established theta function identities using elliptic functions
- Presented additional $q$-trigonometric identities

## Abstract

Finding theta function (or $q$-)analogues for well-known trigonometric identities is an interesting topic. In this paper, we first introduce the definition of $q$-analogues for $\mathrm{tan}z$ and $\mathrm{cot}z$ and then apply the theory of elliptic functions to establish a theta function identity. From this identity we deduce two $q$-trigonometric identities involving $\mathrm{tan}_{q}z$ and $\cot_{q}z,$ which are theta function analogues for two well-known trigonometric identities concerning $\mathrm{tan}z$ and $\cot z.$ Some other $q$-trigonometric identities are also given.