Fourier Series of odd function

I am having trouble with this fourier series:

$\displaystyle f(x)=\left\{\begin{array}{cc}\cos(x),&\mbox{ if }

0\leq x< \pi\\-\cos(x), & \mbox{ if }-\pi\leq x< 0 \end{array}\right.$

I understand that it is an odd function and as such there are no $\displaystyle \cos$ terms, only $\displaystyle \sin$ but i am told that the function should equal

$\displaystyle \frac{8}{\pi} \sum_{m\ =\ 1}^\infty \frac{m}{4m^2-1}\sin(2mx)$

and i am finding it impossible to do so.

Could someone please show me how this is done?