To solve the equation

[
3 \cot(x) - \sqrt{3} \cot(x) \tan(x) = 0
]

we can start by factoring out (\cot(x)):

[
\cot(x) (3 - \sqrt{3} \tan(x)) = 0
]

This gives us two separate equations to solve:

(\cot(x) = 0)(3 - \sqrt{3} \tan(x) = 0)Solving (\cot(x) = 0)

The cotangent function is zero when the tangent function is undefined. This occurs at:

[
x = \frac{\pi}{2} + k\pi \quad \text{for integers } k
]

Solving (3 - \sqrt{3} \tan(x) = 0)

Rearranging gives us:

[
\sqrt{3} \tan(x) = 3 \implies \tan(x) = \frac{3}{\sqrt{3}} = \sqrt{3}
]

The tangent function equals (\sqrt{3}) at:

[
x = \frac{\pi}{3} + n\pi \quad \text{for integers } n
]

Final solutions

Combining results, we have the complete set of solutions:

From (\cot(x) = 0):
[
x = \frac{\pi}{2} + k\pi \quad (k \in \mathbb{Z})
]

From (3 - \sqrt{3} \tan(x) = 0):
[
x = \frac{\pi}{3} + n\pi \quad (n \in \mathbb{Z})
]

Thus, the solutions to the original equation are:

[
x = \frac{\pi}{2} + k\pi \quad \text{and} \quad x = \frac{\pi}{3} + n\pi \quad \text{for integers } k, n.
]