I think I got it but I would still like for someone to confirm. You use want to find the sum on both sides of the previous identity, so you use telescoping sums on the right side and make it sin ( n+1/2)x - sin (1/2)x. You then divide sin(x/2) from both sides and you prove the formula. Correct?
Also, I am unable to do part b. It looks like I just replace the top of the right side of the formula I proved with a different identity but I can't find one that works.