Trigonometric functions (both geometric and circular), relationships between lengths and angles in triangles, and other topics relating to measuring triangles.

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trigonometric sums $ \sum_{k=0}^{n-1}\sin\frac{2k^2\pi}{n}=\frac{\sqrt{n}}{2}\left(\cos\frac{n\pi}{2}-\sin\frac{n\pi}{2}+1\right)$

can someone help me to prove that: $ \sum_{k=0}^{n-1}\sin\frac{2k^2\pi}{n}=\frac{\sqrt{n}}{2}\left(\cos\frac{n\pi}{2}-\sin\frac{n\pi}{2}+1\right)$ ...