Start by using the substitution [tex]\sin 2\theta = 2 \sin \theta \cos \theta[/tex]:
[tex]r^2 = 2 \sin 2\theta[/tex]
[tex]r^2 = 2(2 \sin \theta \cos \theta)[/tex]
[tex]r^2 = 4 \sin \theta \cos \theta[/tex]
Then, multiply both sides by [tex]r^2[/tex]:
[tex]r^4 = 4r^2 \sin \theta \cos \theta[/tex]
[tex](r^2)^2 = 4(r \sin \theta)(r \cos \theta)[/tex]
Since [tex]r^2 = x^2 + y^2[/tex], [tex]r \cos \theta = x[/tex], and [tex]r \sin \theta = y[/tex], we have that
[tex]\bf (x^2 + y^2)^2 = 4xy[/tex]
