math induction of sin(x)-sin(3x)...

how would you use induction to prove this:

$\sin(x)-sin(3x)+sin(5x)-...+(-1)^{(n+1)}sin[(2n-1)x] = \frac{(-1)^{(n+1)}sin2nx}{2cosx}$

I know how you assume its true for n=k, and then prove for n=k+1, but I get to

Left Hand Side: $\frac{(-1)^{(k+1)}sin2kx}{2cosx}+(-1)^{k+2}sin[(2k+1)x]$ but I'm not sure what step to take next.

any help would be appreciated.
Cheers