<s>[INST] How to write $x=2\cos(3t) y=3\sin(2t)$ in rectangular coordinates? How would I write the following in terms of $x$ and $y$? I think I use the inverse $\cos$ or $\sin$?
$$x=2\cos(3t)\,, \quad y=3\sin(2t)$$
 [/INST] We have $\cos(3t)=\frac{x}{2}$.  Using the identity $\cos 2u=2\cos^2(u)-1$, we get
$$\cos(6t)=2\left(\frac{x}{2}\right)^2-1.\tag{1}$$
We also have $\sin(2t)=\frac{y}{3}$. Using the not so well-known identity $\sin(3u)=3\sin u-4\sin^3 u$, we get
$$\sin(6t)=3\left(\frac{y}{3}\right)-4\left(\frac{y}{3}\right)^3.\tag{2}$$
Now use (1) and (2), and the fact that $\cos^2(6t)+\sin^2(6t)=1$ to eliminate $t$. 
We get the remarkably ugly equation
$$\left(  2\left(\frac{x}{2}\right)^2-1  \right)^2+\left( 3\left(\frac{y}{3}\right)-4\left(\frac{y}{3}\right)^3 \right)^2=1.$$
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