The coordinate change from rectangular to polar is
[tex]\begin{cases}x=\rho\cos(\theta)\\y=\rho\sin(\theta)\end{cases}[/tex]
So, you simply have to substitute each occurrence of x and y with their expression:
[tex](x^2+y^2)^2 = x^2-y^2 \iff (\rho^2\cos^2(\theta)+\rho^2\sin^2(\theta))^2 = \rho^2\cos^2(\theta)-\rho^2\sin^2(\theta)[/tex]
Using the fundamental trigonometric equation [tex]\cos(x)^2+\sin(x)^2=1[/tex] we have
[tex](\rho^2)^2 = \rho^2(\cos^2(\theta)-\sin^2(\theta)) \iff \rho^4-\rho^2(\cos^2(\theta)-\sin^2(\theta))=0[/tex]
We can use [tex]\cos(2x)=\cos^2(x)-\sin^2(x)[/tex] to get
[tex]\rho^4-\rho^2(\cos(2\theta))=0 \iff \rho^2(\rho^2-\cos(2\theta))=0[/tex]
