Let $\,\mathbf{x}\longrightarrow\mathbb{R}^2\,$ and $\,\mathbf{y}:J\longrightarrow\mathbb{R}^2\;$ be two parametrizations of a circle, given by
$$\mathbf{x}\left(\theta\right)=\left(\cos\theta,\,\sin\theta\right)\qquad\mathbf{y}\left(t\right)=\left(\frac{1-t^2}{1+t^2},\,\frac{2t}{1+t^2}\right)$$
I am asked to find a relation between both parameters $\,\left(\right.$i.e a dipheomorphism $\,f:I\longrightarrow J\,$ that verifies $\,\mathbf{x}=\mathbf{y}\circ f\left.\right)$
After different attempts, I tried setting $\,\textrm{tg}\left(\frac{\theta}{2}\right)\,$ and it seems to be a right change of variables. However, it was just luck.
Question: Is there a specific method for this kind of exercise?