I am interested in the classical parametrization of rational solutions to $x^2 + y^2 = 1$.

One proof is the classical stereographic projection technique (see, e.g., here): Choose a rational point $P$ on the circle, draw the line $y = 0$ (or a different line), and lines through $P$ match up rational points on the line with rational points on the circle.

A second proof (see, e.g., Elkies) uses Hilbert 90. By this theorem applied to $\mathbb{Q}(i)/\mathbb{Q}$ it is known that any element $x + iy \in \mathbb{Q}(i)$ of norm $1$ is of the form $\frac{a + bi}{a - bi}$ for $a + bi \in \mathbb{Q}(i)$ -- write out what this means, and you're done.

Can these proofs be given a unified treatment, presumably in terms of cohomology? This case of Hilbert 90 can be rephrased as the statement that $H^{1}(\mathbb{Q}(i)/\mathbb{Q}, \mathbb{Q}(i)^{\times}) = 1$, and I was looking for a similarly cohomological interpretation of the stereographic projection map, possibly involving the fact that $x^2 + y^2 = 1$ becomes isomorphic to $zw = 1$ over $\mathbb{Q}(i)$, but I found myself barking up the wrong tree and getting nowhere.