Question 305319
For any a,b we have: 
cos(a) * cos(b) = {{{(1/2)*(cos(a+b) + cos(a-b))}}}
Since a + b = Pi/2, therefore
cos(a) * cos(b) = {{{(1/2)*(cos(Pi/2) + cos(a-b)) = (1/2)*cos(a-b)}}} since cos(Pi/2) = 0.
Since {{{cos(a-b) <= cos(0) = 1}}}, "=" occurs when a = b = Pi/4 = 45 degrees.
Thus, the maximum value of cos(a)*cos(b) is 1.