An easy way to prove these results is to use the well known formula for generating Pythagorean triples and assume that all Pythagorean triples can be generated in the same way. The formula is proved and the ideas discussed at length in the articles Pythagorean Triples I, Pythagorean Triples II and Picturing Pythagorean Triples.

If you want more of a challenge you can try to prove the result from first principles.