1) 3tg2x + √3 = 0
tan 2x = -√3
2x = (2n-1)π - π/3 , where n is an integer
x = nπ - π/6

2) 4sin²x - 12cosx + 3 = 0
4(1 - cos²x) - 12cosx + 3 = 0
4 - 4cos²x - 12cosx + 3 = 0
4cos²x + 12cosx - 1 = 0
cosx = (-12 ± √(12² - 44(-1)))/(2*4)
cosx = (-12 ± √160)/8
cosx = (-12 ± 4√10)/8
cosx = -3/2 ± √10/2
x = arccos(-3/2 ± √10/2)

3) sin x·cos x - √3 cos² x = 0
cosx(sinx - √3 cosx) = 0
cosx = 0 or sinx = √3 cosx
x = arccos(0) or tanx = √3
x = π/2 + nπ or x = π/3 + nπ

4) sin 3x + sin x = 0
3sinx - 4sin³x + 4sinx·cos²x = 0
3sinx - 4sinx(1 - cos²x) + 4sinx(1 - sin²x) = 0
3sinx - 4sinx + 4sinx - 4sin³x + 4sinx - 4sin³x = 0
10sinx - 8sin³x = 0
2sinx(5 - 4sin²x) = 0
sinx(5 - 4sin²x) = 0
sinx = 0 or sin²x = 5/4
sinx = 0 or sin²x = -5/4
x = nπ or x = undefined.