Perhaps
eqns = {1 - 2 Cos[5 w Degree] + 2 Cos[5 x Degree] - 2 Cos[5 y Degree] + 2 Cos[5 z Degree] == 0,
1 - 2 Cos[7 w Degree] + 2 Cos[7 x Degree] - 2 Cos[7 y Degree] + 2 Cos[7 z Degree] == 0,
1 - 2 Cos[11 w Degree] + 2 Cos[11 x Degree] - 2 Cos[11 y Degree] + 2 Cos[11 z Degree] == 0,
1 - 2 Cos[13 w Degree] + 2 Cos[13 x Degree] - 2 Cos[13 y Degree] + 2 Cos[13 z Degree] == 0};
NMinimize[Norm[Map[First, eqns]], {w, x, y, z}]
instantly returns
{3.28831*10^-7, {w -> -10.5456, x -> 32.8669, y -> -30.9045, z -> 16.0925}}