Suppose n and x are fixed. Then (binomial distribution). Also, (negative binomial distribution, see "Alternative formulations"). The idea for the second formula is that the last trial must be a success (n is the minimum number of trials to get x successes), so we need to squeeze x - 1 successes into n - 1 trials.
Now it is easy to prove the required equality if you express the binomial coefficients through factorials.