Super dumb maths question

So i'm watching exam solutions on Parametric equations and Cartesian
equation.

https://www.youtube.com/watch?time_continue=261&v=cHeWFLFODwQ

When converting from Parametric to Cartesian with trig functions, he said; chose an equation that is independent of any other trig function. Im not really sure what he means by this.

Also are Cartesian equations, equations with only x and y?
if not could you please explain to me what makes a Cartesian equation a Cartesian equation and what makes a parametric equation a parametric equation?

-Sorry if this is really obvious. I just want to make sure im not assuming anything.

He chose $\cos(2\theta) = 1-2 \sin^2 (\theta)$ because clearly the LHS is just $y$ and the on the RHS we only have $\sin(\theta)$ which is dependent on $x$. You wouldn't want something like $2\cos^2(\theta) - 1$ on the RHS because this $\cos \theta$ doesn't depend on $y$ nor $x$ so you'd have a hard time (in most cases, not so much here) figuring out how to express it in terms of $x,y$

Yes a Cartesian eq. is one on the usual xy plane that has x and y as the only variables. (for 2-D anyway)

A parametric equation is one that has the main variables of interest dependent on a parameter (or more), which in this case is $\theta$

Parameter on x=2t-1 and y=t^2

is t?