With the following code starting from random initial points

eq = {1 - 2 Cos[5 w Degree] + 2 Cos[5 x Degree] - 2 Cos[5 y Degree] + 
    2 Cos[5 z Degree], 
   1 - 2 Cos[7 w Degree] + 2 Cos[7 x Degree] - 2 Cos[7 y Degree] + 
    2 Cos[7 z Degree], 
   1 - 2 Cos[11 w Degree] + 2 Cos[11 x Degree] - 2 Cos[11 y Degree] + 
    2 Cos[11 z Degree], 
   1 - 2 Cos[13 w Degree] + 2 Cos[13 x Degree] - 2 Cos[13 y Degree] + 
    2 Cos[13 z Degree]};
sol = FindRoot[eq == 0,
   Thread[{{w, x, y, z}, Sort@RandomReal[{0, 90}, 4]}]];
sol
eq /. sol

after a few attempts I found the neat solution {w -> 20., x -> 40., y -> 60., z -> 80.}

NMinimize[{function,constraint}, vars] is shown in the help page.

There does not appear to be a solution where your four equations are approximately equal to zero, but this

NMinimize[
  {Norm[{
      1 - 2 Cos[5 w Degree] + 2 Cos[5 x Degree] - 2 Cos[5 y Degree] + 2 Cos[5 z Degree],
      1 - 2 Cos[7 w Degree] + 2 Cos[7 x Degree] - 2 Cos[7 y Degree] + 2 Cos[7 z Degree],
      1 - 2 Cos[11 w Degree] + 2 Cos[11 x Degree] - 2 Cos[11 y Degree] + 2 Cos[11 z Degree], 
      1 - 2 Cos[13 w Degree] + 2 Cos[13 x Degree] - 2 Cos[13 y Degree] + 2 Cos[13 z Degree]
    }], 
  w < x < y < z < 90 Degree
  }, {w, x, y, z}]

will let you add constraints to your problem.