These results are both true, but to prove them you probably need to know something about complex numbers. For the first problem, is the imaginary part of , where z is an Nth root of unity. The result follows from the fact the sum of the Nth roots of unity is 0. This is a good place to start looking for information on this.
For the second problem, use the trig formula . That will transform the second problem into something looking very much like the first problem (but with cos instead of sin, and 2k instead of k).