It is well known that a part of the unit circle can be rationally parameterized by $$x=\frac{1-t^2}{1+t^2},\,y=\frac{2t}{1+t^2}$$ where $-\infty \lt t \lt \infty$. However, there's no $t\in\mathbb{R}$ which corresponds to the point $(-1,0)$. This led me to the following question:
Do there exist rational functions $f$ and $g$ such that $$x=f(t),\, y=g(t)$$ parameterizes the full unit circle when $t$ varies on a bounded interval?
Let $\cos u=\frac{1-t^2}{1+t^2}$, $\sin u=\frac{2t}{1+t^2}$, then $\cos 2u=...$, $\sin 2u=...$ are example of such rational functions of $t$.