using mi is the way to go.
for n=0, lhs = rhs = 1/x
assume it holds for n
prove it still holds for n+1, using this hint 1/[(x+k)(x+n+1)] = [ 1/(x+k) - 1/(x+n+1) ] / (n+1)
What does it mean to prove something "in the identity"? Do you mean to prove that the first identity implies the second one? That would be surprising because I would expect one can choose is many ways to satisfy the first formula.