find all solutions of equation in [0, 2pi)
2 (sec x)^2 + (tan x)^2 -3 = 0
since sec^2 x = 1+tan^2 x, we have
2(1+tan^2 x)+tan^2 x-3=0
3tan^2 x-1 = 0
tan^2 x = 1/3
tan x = ±1/√3
tan x > 0 in QI, QIII, so
x = π/6, 7π/6
tan x < 0 in QII, QIV, so
x = 5π/6, 11π/6
Actually, since the period of tan x is
π, it would be better to ask for solutions in [0,π) rather than 2π.