isomorphism between quotients of polynomial rings It is fairly known that 
$\mathbb C[x,y,z]/(xy+z^n) \cong \mathbb C[x,y,z]/(z^n+x^2+y^2)$. 
This appears, for example, in the study of singularities of type $A_n$. But, unfortunately, I am not able to prove it. 
How could we establish this isomorphism?
Thanks!

In the left hand side, set $x=u-iv$, $y=u+iv$. Then the LHS is 
$$\mathbb C[u,v, z]/((u^2+v^2)+z^n).$$ 
