[lupire]

> Now the only thing which I'm not sure of is whether there's a way to show that the determinant of a rotation matrix is 1 without assuming the Pythagorean identity already

You can define the determinant that way. Now the question is why the cross multiplication formula for determinant accurately computes the area.

You can prove that via decomposition into right triangles https://youtu.be/_OiMiQGKvvc?si=TyEge1_0W4rb648b

Or you can go in reverse from the coordinate formula, to prove that the area is correctly predicted by the determinant.

[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.