On certain 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 $\tan z$ and $\cot z$ and then apply the theory of elliptic functions to establish a theta function identity. From this identity we deduce two $q$-trigonometric identities involving ${\tan }_{q}z$ and ${\cot }_{q}z,$ which are theta function analogues for two well-known trigonometric identities concerning $\tan z$ and $\cot z.$ Some other $q$-trigonometric identities are also given.

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