On certain q-trigonometric identities

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