Let's say we have the equation $y = \tan(x)$.
We can break this into $y = \frac{\sin(x)}{\cos(x)}$.
Let's say we wanted $a = \sin(x)$.
Therefore, $y = \frac{a}{\sqrt{1-a^2}}$.
However, if I graph the new function y(a) instead of y(x), if looks very different. Why is this?
It looks different because $a$ and $x$ (the things you're plotting on the x asis) are different quantities.
If instead you plotted $y(a(x)) = \frac{\sin(x)}{\sqrt{1-\sin^2(x)}}$ you would get the same graph as $y = \tan(x).$
Also if you plotted $\tan(x)$ versus $\sin(x)$ (instead of vs x) you would get the graph of $\frac{a}{\sqrt{1-a^2}}$