Use the binomial expansion of$(1+i)^{2n}$ to prove that $\binom{2n}{0}-\binom{2n}{2}+\binom{2n}{4}-\binom{2n}{6}+...+(-1)^n\binom{2n}{2n}=2^n\cos {\pi n \over 2}, n\in \mathbb{Z^+}$
I've been trying to solve this problem for quite some time now, but I am unable to make any progress. I first tried to solve it by using De Moivre's theorem which is easy, but not what the question is asking for. Since then I've basically been stuck.