D_n(u)=\frac 1{\sin(u/2)}\left(\frac{\sin(u/2)}2+\sum_{k=1}^n\cos(ku)\sin(u/2)\right)=\frac 1{\sin(u/2)}\left(\frac{\sin(u/2)}2+\sum_{k=1}^n\frac{\sin((k+1/2)u)-\sin((k-1/2)u)}2\right)=\frac{\sin((2n+1)u/2)}{2\sin(u/2)}