Comment by degamad

Yep - I'm just not sure if any of those proofs implicitly assume Pythagoras, and haven't thought through them properly.

I was initially going to say we know that det R = 1 by using the trigonometric identity cos²x+sin²x=1, but then found out that all the proofs of it seem to assume Pythagoras, and in fact, the identity is called the Pythagorean trigonometric identity.