#andy
\[\begin{array}{l}
\cot \left( {x - \frac{\pi }{2}} \right) = \cot (2x + 1)\\
dk:\,\sin \left( {x - \frac{\pi }{2}} \right) \ne 0\\
\sin (2x + 1) \ne 0\\
\Leftrightarrow x \ne \frac{\pi }{2} + k\pi \\
\,\,\,\,\,\,\,x \ne \frac{{k\pi - 1}}{2}\\
\Leftrightarrow x - \frac{\pi }{2} = 2x + 1 + k\pi \\
\Leftrightarrow \,x = \frac{{ - \pi }}{2} - 1 - k\pi
\end{array}\]
