  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.
