ok so there a few problems giving me trouble.

1. Find the critical points of

cos^2 x + cos^2 y constrained to g(x,y) = x+ y = π/4

I used the larangian method and taking partials I managed to get sin 2x = sin 2y. Not sure how to solve that though for x in terms of y or vice versa.

2.

I'm trying to find the minimum of f(x,y) = x^2 + (y-2)^2 constrained to

g(x,y) = x^2 - y^2 = 1

taking the larangian method I got critical points as y = 1 and x = ± sqrt 2

I formed the hessian for f(x,y) and found it is positive defjnite and doesn't rely on the critical points at all. So in this case are both points minimums?

also since there is no maximum, would this contradict the EVT? I said yes because at the least there has to be one maximum on the interval, but I'm not sure if that's right or the right way to explain it.