How to rotate $z=x^2-y^2$ by $45$ degrees about the $z$ axis?

Question

I tried $z=(x\cos(45^{\circ})+y\sin(45^{\circ}))^2-(y\cos(45^{\circ})+x\sin(45^{\circ}))^2$

http://www.wolframalpha.com/input/?i=z%3D(xcos(45)%2Bysin(45))%5E2-(ycos(45)%2Bxsin(45))%5E2

Wolfram says that it is a plane, which is obviously incorrect.

Answer 1

Hint: Axes rotation formula is $$x' = x \cos(\theta) + y \sin(\theta),$$ and $$y' = \color{red}{-}x \sin(\theta) + y \cos(\theta).$$