My question is simple - can all parametric equations be made implicit with enough rearrangement? If so, is there an easy/general method to do so? If not, how would one go about proving that they cannot be made implicit? I am dealing with rather complex trigonometric parametric equations, and I have not succeeded in rearranging them to be implicit.
Edit: in particular, I care about the equations for hypotrochoids following the form: $$x = (R-r)\cos t - a\cos\left(\frac{R-r}{r}t\right)$$$$y = (R-r)\sin t - a\sin\left(\frac{R-r}{r}t\right)$$ where $R$, $r$ and $a$ are variables
In summary, if converting them is not possible, how can I prove they cannot be converted?