KöMaL 2009 A474

The point $Q$ is located in the convex $n$-gon $P_1P_2\ldots P_n$. Show that $\sum_{k=1}^n \big(\cot QP_{k-1}P_k\sphericalangle + \cot QP_{k+1}P_k\sphericalangle\big) \cdot \overrightarrow{QP_k} = 0.$ (The indices are considered modulo $n$, e.g. $P_0$=$P_n$ and $P_{n+1}$=$P_1$.)