Recall that the tangent function is defined by
tan(x) = sin(x)/cos(x)
Also recall the double angle identity for sine,
sin(2x) = 2 sin(x) cos(x)
Then the equation is the same as
3 sin(x)/cos(x) = 4 sin(x) cos(x)
Move everything to one side to prepare to factorize:
3 sin(x)/cos(x) - 4 sin(x) cos(x) = 0
sin(x)/cos(x) (3 - 4 cos²(x)) = 0
As long as cos(x) ≠ 0, we can omit the term in the denominator, so we're left with
sin(x) (3 - 4 cos²(x)) = 0
and so
sin(x) = 0 or 3 - 4 cos²(x) = 0
sin(x) = 0 or cos²(x) = 3/4
sin(x) = 0 or cos(x) = ±√3/2
On the interval [0, 2π),
• sin(x) = 0 for x = 0 and x = π
• cos(x) = √3/2 for x = π/6 and x = 11π/6
• cos(x) = -√3/2 for x = 5π/6 and x = 7π/6
(None of these x make cos(x) = 0, so we don't have to omit any extraneous solutions.)