Answer
$x^2+y^2=(1-x)^2$
Work Step by Step
After multiplying with the denominator of the fraction: $r(1+\cos{\theta})=1\\r+r\cos{\theta}=1\\r=1-r\cos{\theta}\\r^2=(1-r\cos{\theta})^2$.
We know that $r^2=x^2+y^2$ and that $x=r\cos\theta,y=r\sin\theta$.
Hence $r^2=(1-r\cos{\theta})^2$ becomes: $x^2+y^2=(1-x)^2$