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.

It appears that the Norm of your function is minimized when w==0 Degree and x,y,z are as close to 90 Degree as possible. You can verify this by substituting various values of w,x,y,z. It might be possible that this is only a local minimum and there is one or more smaller minima elsewhere.