Form of rational solutions to $a^2+b^2=1$? Is there a way to determine the form of all rational solutions to the equation $a^2+b^2=1$?

I up-voted yunone's answer, but I notice that it begins by saying "if you know some field theory", and then gets into $N_{\mathbb{Q}(i)/\mathbb{Q}}(a_bi)$, and then talks about Galois groups, and then Hilbert's Theorem 90, and tau functions (where "$\tau$ is just the complex conjugation map in this case" (emphasis mine)).
Sigh.
I would have no hesitation about telling a class of secondary-school pupils that all Pythagorean triples are of the form
$$
(a,b,c) = (m^2-n^2,2mn,m^2+n^2)
$$
and that they are primitive precisely if $m$ and $n$ are coprime and not both odd.
That means rational points on the circle are
$$
\left(\frac a c, \frac b c\right) = \left( \frac{m^2-n^2}{m^2+n^2}, \frac{2mn}{m^2+n^2} \right)
$$
and they're in lowest terms precisely under those circumstances.
But how much of what appears in my first paragraph above would I tell secondary-school pupils?  Maybe it's better not to lose the audience before answering the question.
