Answer by Firestone for Proofs of theorems that proved more or deeper results than what was first supposed or stated as the corresponding theorem

I hesitate to offer an example from two millennia earlier than you have requested, but perhaps it may qualify as surely having passed unnoticed by many thousands of students. Euclid's proof of the Pythagorean theorem (his 1.47) shows not only that the big square is the sum of the smaller squares, but also how it is divided into parts equal to those squares.