差分

:$$\displaystyle S_{n}=\frac{\displaystyle\left(\cos\theta+i\sqrt{\cos^2sin\theta-1}\right)^n-\left(\cos\theta-i\sqrt{\cos^2sin\theta-1}\right)^n}{\displaystyle2displaystyle2i\sqrt{\cos^2sin\theta-1}}\left(=\frac{\sin n\theta}{\sin\theta}\right)=\sum_{k=0}^{\lfloor (n-1)/2\rfloor}\binom{n-k-1}{k}(-1)^k(2\cos\theta)^{n-2k-1}$$