$\begin{array}{l}
a)\,\,\,{\tan ^2}\left( {\frac{\pi }{2}} \right) - \left( {1 + \sqrt 3 } \right)\cot x + \sqrt 3 = 0\\
\Leftrightarrow {\cot ^2}x - \left( {1 + \sqrt 3 } \right)\cot x + \sqrt 3 = 0\\
\Leftrightarrow \left[ \begin{array}{l}
\cot x = 1\\
\cot x = \sqrt 3
\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}
x = \frac{\pi }{4} + k2\pi \\
x = \frac{\pi }{6} + k\pi
\end{array} \right.\\
b)\,\,\,\sqrt 3 {\cot ^2}x + \left( {\sqrt 3 - 1} \right)\tan \left( {\frac{{3\pi }}{2} - x} \right) - 1 = 0\\
\Leftrightarrow \sqrt 3 {\cot ^2}x + \left( {\sqrt 3 - 1} \right)\cot x - 1 = 0\\
\Leftrightarrow \left[ \begin{array}{l}
\cot x = - 1\\
\cot x = \frac{1}{{\sqrt 3 }}
\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}
x = \frac{{ - \pi }}{4} + k\pi \\
x = \frac{\pi }{3} + k\pi
\end{array} \right.
\end{array}$
