The answer is [tex] 2xy = 1 [/tex]
The transformation between polar and cartesian coordinates is
[tex] r = \sqrt{x^2+y^2},\qquad r\sin(\theta)=y,\qquad r\cos(\theta)=x [/tex]
Now use the trigonometric identity [tex] \sin(2\theta)=2\sin(\theta)\cos(\theta) [/tex] to rewrite the expression as
[tex] r^2\sin(2\theta) = 2r^2\sin(\theta)\cos(\theta) = 2r\sin(\theta)r\cos(\theta) [/tex]
Now recall the transformations to rewrite
[tex] 2r\sin(\theta)r\cos(\theta) = 2xy [/tex]
