$\begin{array}{l}
\frac{3}{{{{\sin }^2}x}} + \left( {6 + \sqrt 3 } \right)\cot x + 2\sqrt 3 - 3 = 0\\
\Leftrightarrow 3\left( {1 + {{\cot }^2}x} \right) + \left( {6 + \sqrt 3 } \right)\cot x + 2\sqrt 3 - 3 = 0\\
\Leftrightarrow 3{\cot ^2}x + \left( {6 + \sqrt 3 } \right)\cot x + 2\sqrt 3 = 0\\
\Leftrightarrow \left[ \begin{array}{l}
\cot x = - \frac{{\sqrt 3 }}{3}\\
\cot x = - 2
\end{array} \right.\\
\Leftrightarrow \left[ \begin{array}{l}
x = - \frac{\pi }{3} + k\pi \\
x = {\mathop{\rm arccot}\nolimits} \left( { - 2} \right) + k\pi
\end{array} \right.
\end{array}$
