报告题目 (Title)：Trigonometric identities and quadratic residues
(三角恒等式和二次剩余)
报告人 (Speaker)：孙智伟教授（南京大学）
报告时间 (Time)：2022年8月10日(周三) 15:00
报告地点 (Place)：腾讯会议 会议ID 432427242 密码220810
邀请人(Inviter)：陈旦旦
主办部门：理学院数学系
报告摘要：
In this talk, we introduce some new identities for trigonometric functions, some of which involve quadratic residues. For example, we determine the values of $\prod_{k=1}^{(p-1) / 2}\left(1+\tan \pi \frac{k^{2}}{p}\right)$ and $\prod_{k=1}^{(p-1) / 2}\left(1+\cot \pi \frac{k^{2}}{p}\right)$ for any odd prime $p$, and prove (joint with Hao Pan) the novel identity $$\sum_{k=1}^{n}(-1)^{k}(\cot k x) \sin k(n-k) x=\frac{1-n}{2}$$ for any positive odd integer $n$.