<s>[INST] Convert $r^2= 9 \cos 2 \theta$ into a Cartesian equation This is how I tried so far...
$r^2= 9 \cos 2 ( \theta)$
$\cos (2 \theta) = \cos ^2 (\theta) - \sin^2 (\theta)$  and 
$r^2= x^2 + y^2$
so, it will become
$x^2 + y^2 = 9 [\cos^2 (\theta) - \sin^2 (\theta) ]$
 [/INST] Starting with 
$$r^2=9\cos2\theta$$
Applying the double angle identity
$$r^2=9\cos^2\theta-9\sin^2\theta$$
Now multiply both sides by $r^2$, apply $x=r\cos\theta,y=r\sin\theta$ and $x^2+y^2=r^2$ to obtain
$$(x^2+y^2)^2=9x^2-9y^2$$
Which, when expanded gives
$${x}^{4}+2\,{x}^{2}{y}^{2}+{y}^{4}-9\,{x}^{2}+9\,{y}^{2}=0$$
</s>