Undo this transformation I have two variables $x$, $y$ and calculate the following:
$a = \frac{x}{\sqrt{x^2+y^2}}$, $b = \frac{y}{\sqrt{x^2+y^2}}$
Using $a$ and $b$ is there a way I can derive my original $x$ and $y$?

By squaring we get $$a^2=\frac{x^2}{x^2+y^2},b^2=\frac{y^2}{x^2+y^2}$$ and we get
$$a^2+b^2=\frac{x^2+y^2}{x^2+y^2}=1$$
and you can not compute $x$ or $y$.
