$\sum_{k=1}^n\sin kx=\frac{\cos\left(\frac{1}2x\right)-\cos\left(n+\frac{1}2\right)x}{2\sin(x/2)},x\neq0,\pm\pi,\pm 2\pi,\ldots$
How can we sum up $\sin$ and $\cos$ series when the angles are in arithmetic progression?