## Calculus (3rd Edition)

$$(x^2+y^2)^2=x^2-y^2.$$
Since $r^2=\cos (2\theta)$, then we have $$r^2=\cos^2\theta-\sin^2\theta=\frac{x^2}{r^2}-\frac{y^2}{r^2}.$$ Hence, $r^4=x^2-y^2$, which can be written in the form $$(x^2+y^2)^2=x^2-y^2.$$