Abstract In this paper we will give q-analogues for the Pythagorean trigonometric identity sin 2 ⁡ z + cos 2 ⁡ z = 1 {\sin^{2}z+\cos^{2}z=1} in terms of Gosper’s q-trigonometry. We shall also give new q-analogues for the duplicate trigonometric identity sin ⁡ ( x - y ) ⁢ sin ⁡ ( x + y ) = sin 2 ⁡ x - sin 2 ⁡ y {\sin(x-y)\sin(x+y)=\sin^{2}x-\sin^{2}y} . Moreover, we shall give a short proof for an identity of Gosper, which was also established by Mező. The main argument of our proofs is the residue theorem applied to elliptic functions.

中文翻译：

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关于三角恒等式的q模拟

摘要在本文中，我们将根据勾勒的q-给出毕达哥拉斯三角恒等式sin 2⁡z + cos 2⁡z = 1 {\ sin ^ {2} z + \ cos ^ {2} z = 1}的q类比。三角学。我们还将为重复的三角恒等式sin⁡（x-y）⁢sin⁡（x + y）= sin 2⁡x-sin 2⁡y {\ sin（xy）\ sin（x + y） ）= \ sin ^ {2} x- \ sin ^ {2} y}。此外，我们将简短说明Gosper的身份，该身份也是Mező建立的。我们的证明的主要论据是适用于椭圆函数的残差定理。