tan(x)+sec(x)=2cos(x),
(sin(x)+1)/cos(x)=2cos(x),
sin(x)+1=2cos2(x)=2-2sin2(x),
2sin2(x)+sin(x)-1=0=(2sin(x)-1)(sin(x)+1).
Therefore, sin(x)=½, x=30° (or π/6 radians) and sin(x)=-1, x=270° (3π/2 radians). However, tan(270°) and sec(270°) cannot be defined so cannot be solutions of the original equation. That means there's only one main angle, 30° (π/6). Its supplement is also valid 180-30=150° (5π/6).