How to find the Cartesian equation of a plane curve from a parametric equation? More specifically, how to express 
$$\begin{aligned}x(t) &=\frac{2t}{1+t^2}\\
y(t) &=\frac{1-t^2}{1+t^2}\end{aligned}$$
in terms of $x$ and $y$? I attempted adding the two, getting a square from the numerator and a few other methods, but running out of time.
EDIT: please don't work backwards, and try to show this as simple as possible. Overcomplication or overdefining equations may confuse me even more.

Those are the parametric equations of the unit circle $x^2+y^2=1$. 
In fact
$$x^2+y^2=\frac{(2t)^2+(1-t^2)^2}{(1+t^2)^2}=\frac{4t^2+1-2t^2+t^4}{(1+t^2)^2}=\frac{1+2t^2+t^4}{(1+t^2)^2}=1.$$
