How to contribute to friend's mathematics paper without overshadowing?
Tell him what you have figured out.  Maybe he will want to modify his own proof and add you as an author.  Maybe he will just add a note that it has been pointed out to him that the result can also be proven using other specified methods.  Maybe he will look over your shorter proof with interest but not change his manuscript at all.

Understanding how a result can be proved is obviously of key importance in mathematics, but do not forget that result that he has proved ought to have some interest unto itself.  It can be more important to know that something is true than to have the most elegant proof of its truth, and realizing that a result is provable and interesting is a significant accomplishment on its own, even if a "better" proof can be found later.