Given $$y^2 = x^2 - x^4,$$
how do you represent this in polar form?
I tried substituting $$x=r\cos \theta$$ and $$ y= r\cos \theta$$ which gave me $$r = \sqrt{\frac {\cos^2 \theta - \sin^2 \theta} {\cos^4 \theta}} $$
However, the square root does not give a real number e.g. when $$\theta=1$$ $r$ is undefined.
What would be the correct equation then?
Perhaps a plot makes clear: Your equation defines a closed curve for $x\in(-1,1)$
Since |x| is bounded by 1, the replacement in polar coordinates yields
$$ \text{ParametricPlot[} \{\sin[ \phi], \cos[\phi] \sin[\phi\}, \{ \phi, -\pi, \pi \}]$$
same picture. The height of $1/2$ and the double loop is due to
$$\cos \phi\ \sin \phi = \frac{1}{2}\ \sin (2\phi)$$