Question 38
(HUN 5) Let $r$ and $m$ $(r\le m)$ be natural numbers and $A_k=\frac{2k-1}{2m}\pi$. Evaluate
$$\frac1{m^2}\sum_{k=1}^m\sum_{l=1}^m\sin(rA_k)\sin(rA_l)\cos(rA_k-rA_l).$$
$$\frac1{m^2}\sum_{k=1}^m\sum_{l=1}^m\sin(rA_k)\sin(rA_l)\cos(rA_k-rA_l).$$