The maximum value of the cosine is 1.
Each of the cosines should be 1 for that value. By symmetry, at that point.
The problem is the following:
max(cos(xy)+cos(xz)+cos(yz)), where
a) x+y+z=180 degrees (or Pi);
b) x,y,z are the angles of a triangle.
Problem should be solved WITHOUT using calculus (derivatives etc.). One can use inequalities, properties of trigonometrical functions etc.
Please help to solve this problem!!!