Answer
$(x^2+y^2)^2=4x^2-4y^2$.
Work Step by Step
To convert from polar to rectangular coordinates we use $x=r\cos\theta$ and $y=r\sin\theta$. To convert from rectangular to polar coordinates, we use $r=\pm\sqrt{x^2+y^2}$, $\tan\theta=\frac{y}{x}$, but aware of which quadrant the point lies in.
Hence here $r^2=4\cos2\theta=4\cos^2\theta-4\sin^2\theta\\r^4=4r^2\cos^2\theta-4r^2\sin^2\theta\\(x^2+y^2)^2=4x^2-4y^2$.