Having multiplied throughout by $\displaystyle 2\sin(x/2),$ consider a general term in the resulting series, $\displaystyle 2\cos(kx)\sin(x/2).$
Use the trig identity to break this into two terms.
Apply this to each term of your series in turn, when you've done this, you should find that there is a whole lot of cancellation, after which you need to use the trig identity again.
Don't forget to divide the resulting expression by $\displaystyle 2\sin(x/2),$ .