Calculating equations for image of variety under map

Question

This is another qualifying exam practice problem that I'm having trouble with. I'm asked to compute polynomials which cut out the image of the variety $x^2+y^2-1=0$ in $\mathbb{C}^2$ under the map $\mathbb{C}^2 \rightarrow \mathbb{C}^2$ defined by $(x,y) \mapsto (xy,x^2-y^2)$. It's clear that one equation that is needed is $(xy^2+(x^2-y^2)^2-1=0$, but I have no idea how to show that more equations are not needed. Thanks for any help!