Why does maxima occur mostly at equality with a fixed condition for geometrical problems like in sum of sines, sum of cosines (considering a triangle), and also in problems like finding maximum area for a rectangle of fixed perimeter? What is the reason behind this equality in solutions to maxima?
For example: $\cos a + \cos b + \cos c \le 3/2$ when $a=b=c$ for a triangle.
Edit:
I mean why does mathematics always try to make things equal and I have observed that as we increase constraints the equality begins to diminish. Example: the Best Box this question has two consraints that is fixed edge length and fixed surface... so we just have one dimension of equality... (only base is square) had there been just one constraint, the answer would have been a cube... i.e. two dimensions of equality...? What is the reason that everything tries to equalize for maximisation?
The primary reason is because multiplication as a function of (x,y) increases faster than addition does, and both are commutative.
In all the problems you have listed, there is a sum of some sort in the constraints. In the function to be optimised, there is a product. This means that for the function to be at a maximum, it should utilise operations that increase the operands to a greater extent.
Ask yourself this; for any two variables x,y related by a constant c where x+y=c, how would one maximise values of x and y?
Likewise, for xy=c how would one maximise x and y for the function f(x,y)= x^y+y^x? This is a similar case where a commutative function requires maximisation with respect to a commutative constraint.
If all variables are treated the same by the function, i.e. if you swapped width and height around or if you exchanged theta with alpha in a right angled triangle and all formulas are identical then why should the function maximise them any differently?