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.}

Thanks a lot Bill Simpson, there is a bit of problem occurring , that is the values of x,y & z are appearing same. they are not changing.