Expand (1-x)^n and sub in x=1 for both that and (1+x)^n, finally add the 2 together and then continue to manipulate the eqn to show whatever needs to be proven
Substitute into the binomial equation call call it 1. Substitute into another binomial equation call call it 2. Add the 2 equations. Noting when you add an even number + odd number = odd number. The 'odd' binomial terms cancel and the 'even' terms are doubled. Hope that is a starting point.
I am writing a full Latex solution so I'll post that soon.