[2016年早稲田大学商学部・数学第1問]3角関数の総和

\begin{align*} \left(\sum_{k=1}^{2016}k\sin\frac{\left(2k-1\right)\pi}{2016}\right)\sin\frac{\pi}{2016} & =\left(\sum_{k=1}^{1008}k\sin\frac{\left(2k-1\right)\pi}{2016}+\sum_{k=1009}^{2016}k\sin\frac{\left(2k-1\right)\pi}{2016}\right)\sin\frac{\pi}{2016}\\ & =\left(\sum_{k=1}^{1008}k\sin\frac{\left(2k-1\right)\pi}{2016}+\sum_{k=1}^{1008}\left(k+1008\right)\sin\frac{\left(2\left(k+1008\right)-1\right)\pi}{2016}\right)\sin\frac{\pi}{2016}\\ & =\left(\sum_{k=1}^{1008}k\sin\frac{\left(2k-1\right)\pi}{2016}+\sum_{k=1}^{1008}\left(k+1008\right)\sin\frac{\left(2k+2016-1\right)\pi}{2016}\right)\sin\frac{\pi}{2016}\\ & =\left(\sum_{k=1}^{1008}k\sin\frac{\left(2k-1\right)\pi}{2016}+\sum_{k=1}^{1008}\left(k+1008\right)\sin\left(\frac{\left(2k-1\right)\pi}{2016}+\pi\right)\right)\sin\frac{\pi}{2016}\\ & =\left(\sum_{k=1}^{1008}k\sin\frac{\left(2k-1\right)\pi}{2016}-\sum_{k=1}^{1008}\left(k+1008\right)\sin\left(\frac{\left(2k-1\right)\pi}{2016}\right)\right)\sin\frac{\pi}{2016}\\ & =\left(\sum_{k=1}^{1008}k\sin\frac{\left(2k-1\right)\pi}{2016}-\sum_{k=1}^{1008}k\sin\left(\frac{\left(2k-1\right)\pi}{2016}\right)-1008\sum_{k=1}^{1008}\sin\left(\frac{\left(2k-1\right)\pi}{2016}\right)\right)\sin\frac{\pi}{2016}\\ & =-1008\left(\sum_{k=1}^{1008}\sin\left(\frac{\left(2k-1\right)\pi}{2016}\right)\right)\sin\frac{\pi}{2016}\\ & =-1008\sum_{k=1}^{1008}\left\{ \sin\left(\frac{\left(2k-1\right)\pi}{2016}\right)\sin\frac{\pi}{2016}\right\} \\ & =-1008\cdot\frac{-1}{2}\sum_{k=1}^{1008}\left\{ \cos\left(\frac{\left(2k-1\right)\pi}{2016}+\frac{\pi}{2016}\right)-\cos\left(\frac{\left(2k-1\right)\pi}{2016}-\frac{\pi}{2016}\right)\right\} \cmt{\because\sin\alpha\sin\beta=-\frac{1}{2}\left(\cos\left(\alpha+\beta\right)-\cos\left(\alpha-\beta\right)\right)}\\ & =-1008\cdot\frac{-1}{2}\sum_{k=1}^{1008}\left\{ \cos\left(\frac{k\pi}{1008}\right)-\cos\left(\frac{\left(k-1\right)\pi}{1008}\right)\right\} \\ & =-1008\cdot\frac{-1}{2}\left\{ \cos\left(\frac{1008\pi}{1008}\right)-\cos\left(\frac{\left(1-1\right)\pi}{1008}\right)\right\} \\ & =-1008\cdot\frac{-1}{2}\left(\cos\pi-\cos0\right)\\ & =-1008\cdot\frac{-1}{2}\left(-1-1\right)\\ & =-1008 \end{align*}