Have i misunderstood the question? I was reading it as the polar co-ords were parametrised by (r,x) instead of (r,theta)
To finish off from my post use
x=rcosx y=rsinx ; x and y on LHS are the cartesian co-ordinates, the r and x are the polar co-ords.
If not then there is nothing complicated to do.
Since r^2=x^2+y^2
r^2 = sec(2x) gives x^2+y^2=sec2x
r^2 = cosec(2x) gives x^2+y^2=cosec(2x)