Finding a Function $\phi$ Given a Parametric Equation

Question

I am trying to find a function $\phi$ that satisfies the equation $\phi(\psi(t)) = t$, where $\psi(t)$ is given by the following parametric equation:

$\psi(t) = \left(\frac{t^2-1}{1+t^2},\frac{2t}{1+t^2}\right)$

I have tried different methods, but I am struggling to find a suitable function $\phi$. I want to show that rational parametrization of curve $x^2+y^2=1$ (function $\psi$) is valid. Can anyone help me with this problem?

Thank you!

Answer 1

Solution: $$\phi(x,y)=\frac y{1-x}.$$