Minimize[{x^2 - y^2,
Cos[x - y] >= 1/2, -5 <= x <= 5, -5 <= y <= 5}, {x, y}] (* returns unevaluated *)
Minimize[{x^2 - y^2,
Reduce[{Cos[x - y] >= 1/2, -5 <= x <= 5, -5 <= y <= 5}, {x,
y}]}, {x, y}] (* returns *)
{1/9 (-150 \[Pi] + 25 \[Pi]^2), {x -> 5 - (5 \[Pi])/3, y -> 5}}
