When expanding (x+y)^n + (x-y)^n think of the second part instead as (x+(-y))^n. Once you expand the two equations with the binomial theorem you will have two expanded formulas where the only difference is one has y^k and the other has -y^k. This means that when k is odd the two will cancel out, and when k is even the two add together basically giving you the expansion of 2*(x+y)^n (for only even values of k).
