This is too complicated. Prove that

nC0 + nC1 + nC2 + nC3 + ... + nCn = 2^n

and

nC0 - nC1 + nC2 - nC3 + ... + nCn = 0 (assuming n is even).

Then add and subtract these two identities. They follow from 1+1 = 2 and 1-1 = 0 and the Binomial Theorem.