√3cot(3x) + 1 = 0

cot(3x) = -1/√3

3x = 2pi/3 or 3x = 5pi/3 or 3x = 8pi/3 etc.

x = 2pi/9 or x = 5pi/9 or x = 8pi/9 etc.

Since 0 <= x <= 2pi is specified,

x = 2pi/9 or x = 5pi/9 or x = 8pi/9 or x = 11pi/9 or x = 14pi/9 or x = 17pi/9

cot (3x) = - 1 / √3

1 / tan (3x) = - 1 / √3

tan (3x) = - √3

3x = 2π/3 , 5π/3 , 8π/3 , 11π/3 , 14π/3 , 17π/3

x = 2π/9 , 5π/9 , 8π/9 , 11π/9 , 14π/9 , 17π/9

√3 cot(3x) + 1 = 0

√3 cot(3x) = −1

cot(3x) = −1/√3

tan(3x) = −√3

3x = 2π/3 + πk = (2+3k)π/3

x = (2+3k)π/9

For 0 ≤ x ≤ 2π

x = 2π/9, 5π/9, 8π/9, 11π/9, 14π/9, 17π/9

However, I'm not sure I understand why you have a question.

YOU seem to think that 17π/9 is a solution, but then you say that YOU keep getting 2π/9, 5π/9, 8π/9, 11π/9, 14π/9.

If you know that 17π/9 is a solution, and it fits the pattern, and is within the given interval, why the confusion?