Answer
$x^{2}+2y-1=0$
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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$r=\displaystyle \frac{1}{1+\sin\theta}\qquad/\times(1+\sin\theta)$
$ r+r\sin\theta=1\qquad$... subsitute: $r\sin\theta=y$
$r+y=1$
$ r=1-x\qquad$... square both sides
$ r^{2}=(1-y)^{2}\qquad$...substitute $r^{2}$
$x^{2}+y^{2}=(1-y)^{2}$
$x^{2}+y^{2}=1-2y+y^{2}$
... move all from RHS to LHS
$x^{2}+2y-1=0$
