I'll illustrate the problem I'm trying to solve with an example.
Let's consider the equations
$$ x = \cos (t) $$ $$ y = \sin (t) $$
We know that these are a parametric form of the unit circle.
In this case, we can actually prove that without plotting because if we square and add the two equations, we get
$$ x^2 + y^2 = \cos^2 (t) + \sin^2 (t) = 1 $$
which is an equation in $x, y$ and describes the circle of radius $1$. But we didn't do it by some formal method, just by recognition.
I know that parametric equations that describe straight lines can easily be solved by substitution. For example,
$$ x = 2t $$ $$ y = 4t + 2 $$ $$ \implies y = 2x + 2 $$
But what other methods exist of generally finding an algebraic equation if we are given parametric form of a curve?
The example that I'm particularly struck at is
$$ x = t\cos(t) $$ $$ y = t\sin(t) $$
But in general, what is the strategy to solve any given parametric curve's general/cartesian form?