Question

If cos x+cos y+cos z=sin x+sin y+sin z then the possible

Answer 1

There could be numerous solutions to the given problem. As x ,  y and z are not related, we cannot derive a simple formula using algebraic/trigonometric methods.

Cos x + cos y + cos z = sin x + sin y + sin z
=> Cos x - sin x + cos y - sin y + cos z - sin z = 0
=> √2 (sinπ/4 cos x - cosπ/4 sin x) + √2 (sinπ/4 cos y - cos π/4 sinx) 
       + √2 (sinπ/4 cos z - cosπ/4 sin z) = 0
=> Sin (π/4 - x) + sin(π/4 - y) + Sin (π/4 - z) = 0
=> 2 Sin [π/8  - (x+y)/2] Cos (x-y)/2 = Sin (z - π/4)

There could be many solutions,
One solution is:  x = y =  z = π/4

Some more solutions:  z = π/4,  x-y = π
                                  z = π/4,  x+y = π/4
                                  z = π/4,  x+y = -7π/8
                                  z = π/4,  x+y = 9π/8