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