Let $F$ be a field of characteristic $\neq 2$ and let $u$ be transcendental over $F$. Suppose $u^2+v^2=1$. Show that $F(u,v)$ is a purely transcendental extension by showing that $F(u,v) = F(\frac{1+v}{u})$.

At first glance, my goal is to prove that $\frac{v}{u} \in F(\frac{1+v}{u})$ but after several attempts, I still couldn't get it right. Any hint would be appreciated! Thanks.