Putnam 1987 B3

Let $F$ be a field in which $1+1 \neq 0$. Show that the set of solutions to the equation $x^2+y^2=1$ with $x$ and $y$ in $F$ is given by $(x,y)=(1,0)$ and $$(x,y) = \left( \frac{r^2-1}{r^2+1}, \frac{2r}{r^2+1} \right)$$ where $r$ runs through the elements of $F$ such that $r^2\neq -1$.