We have \begin{equation} z w = \frac{(x + i y) (x - i y)}{x^2 + y^2} = \frac{(x^2 - i x y + i x y - i^2 y^2)}{x^2 + y^2} = \frac{x^2 + y^2}{x^2 + y^2} = 1 \end{equation} and an equivalent result holds for $z w$. $\square$