$\begin{array}{l}
\dfrac{{\tan x}}{{1 - {{\tan }^2}x}} = \dfrac{1}{2}\cot \left( {x + \dfrac{\pi }{4}} \right)\\
\Leftrightarrow \dfrac{{2\tan x}}{{1 - {{\tan }^2}x}} = \cot \left( {x + \dfrac{\pi }{4}} \right)\\
\Leftrightarrow \tan 2x = \cot \left( {x + \dfrac{\pi }{4}} \right)\\
\Leftrightarrow \cot \left( {\dfrac{\pi }{2} - 2x} \right) = \cot \left( {x + \dfrac{\pi }{4}} \right)\\
\Leftrightarrow x + \dfrac{\pi }{4} = \dfrac{\pi }{2} - 2x + k\pi \\
\Leftrightarrow 3x = \dfrac{\pi }{4} + k\pi \\
\Leftrightarrow x = \dfrac{\pi }{{12}} + \dfrac{{k\pi }}{3}\left( {k \in \mathbb{Z}} \right)
\end{array}$
