Answer
$x^{4}+2x^{2}y^{2}+y^{4}-2xy=0$
Work Step by Step
Formulas relating rectangular $(x,y)$ and polar $(r,\theta)$ coordinates:
$ x=r\cos\theta$ and $ y=r\sin\theta$
$r^{2}=x^{2}+y^{2}$ and $\displaystyle \tan\theta=\frac{y}{x}$.
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Apply the double angle identity on the RHS:
$ r^{2}=2\sin\theta\cos\theta$
Multiplying both sides with $r^{2}$ gives
$r^{2}\cdot r^{2}=2(r\sin\theta)(r\cos\theta)$
apply the above formulas to substitute $r^{2},\ r\sin\theta$ and $ r\cos\theta$
$(x^{2}+y^{2})^{2}=2xy$
$x^{4}+2x^{2}y^{2}+y^{4}-2xy=0$
