Gosper in 2001 introduced the q-trigonometric functions and conjectured many interesting q-trigonometric identities. In this paper, we apply Riemann’s addition formula to deduce two Jacobi theta function identities. From these theta function identities, we confirm a q-trigonometric identity conjectured by Gosper and establish two other similar results. As an application, two theta function analogues for Ptolemy’s theorem are given.

中文翻译：

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Gosper的aq-三角恒等式的证明

戈斯珀（Gosper）在2001年引入了q三角函数，并推测出许多有趣的q三角恒等式。在本文中，我们使用黎曼加法公式来推导两个Jacobi theta函数恒等式。从这些theta函数恒等式中，我们确定了Gosper猜想的q三角恒等式，并建立了另外两个相似的结果。作为应用，给出了托勒密定理的两个theta函数类似物。