The second identity can be proven by first recognizing that
(x+y)^n=\sum_{k=0}^{n} x^{k} y^{n-k} \left _n C _k
then use the fact that
(x+y)^{a+b}=(x+y)^{a} (x+y)^{b}
Expand the terms using the summation form, then compare like terms to obtain your identity.