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.