Left a name in Excalibur!

There is a fairly easy question in the problem corner of the latest mathematical excalibur:
Show that $\displaystyle \sum_{k=0}^{n-1}(-1)^k\cos^n\left(\frac{k\pi}{n}\right)=\frac{n}{2^{n-1}}$, for any positive integer $n$.

The term  $2^{n-1}$ is reminiscent of something, isn't it? What is it? How it works?

But I guess there must be some combinatorics/probabilistic argument, especially the use of inclusion-exclusion principle.

I am glad that Kin  Li adopted my solution of the strange inequality and quoted my name^^.
For details:
http://www.math.ust.hk/excalibur/v14_n3.pdf
(You can search LEE Ching Cheong to see where I am XD)