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 $\mathrm{tan}z$ 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.