Critical points : $ z = \cos^2x + \cos^2y$ for $ y-x = \frac{\pi}{4}$ Find and classify the critic points (maximum, minimum or neither) of the function $$z=z(x,y) = \cos^2x + \cos^2y $$
if $y-x = \large\frac{\pi}{4}$
I've find $x = -\large\frac{\pi}{8} + \large\frac{k \pi}{2}$ and $y = \large\frac{\pi}{8} + \large\frac{k \pi}{2}$ ($k \in \Bbb{Z}$), but I sm having troubles to classify the points.
Thanks for your help!

Use $$\cos^2x+\cos^2y=\frac{1+\cos2x+1+\cos2y}2=1+\frac{\cos2x+\cos2\left(x+\frac\pi4\right)}2$$
$$=1+\frac{\cos2x-\sin2x}2$$  as $\cos\left(\frac\pi2+y\right)=-\sin y$
So, the problem reduces to single variable
