Answer
$r^2=\dfrac{1}{1+\sin \theta \cos \theta}$
Work Step by Step
Conversion of polar coordinates and Cartesian coordinates are as follows:
a) $r^2=x^2+y^2 \implies r=\sqrt {x^2+y^2}$
b) $\tan \theta =\dfrac{y}{x} \implies \theta =\tan^{-1} (\dfrac{y}{x})$
c) $x=r \cos \theta$
d) $y=r \sin \theta$
Since, we have $x=r \cos \theta$ and $y=r \sin \theta$ and $r^2=x^2+y^2 \implies r=\sqrt {x^2+y^2}$
Thus, we have an equivalent polar equation $r^2+r \cos \theta (r \sin \theta)=1$ or, $r^2+r^2 \cos \theta \sin \theta=1$
Or: $r^2=\dfrac{1}{1+\sin \theta \cos \theta}$
