$\begin{array}{l}
\cot \left( {\cos x} \right) = \tan \left( {\dfrac{\pi }{2} - \sin x} \right)\\
\Leftrightarrow \cot \left( {\cos x} \right) = \cot \left( {\sin x} \right)\\
\Leftrightarrow \cos x = \sin x + k\pi \\
\Leftrightarrow \sqrt 2 \cos \left( {x + \dfrac{\pi }{4}} \right) = k\pi \\
\Rightarrow - \sqrt 2 \le k\pi \le \sqrt 2 \Rightarrow k = 0\\
\Rightarrow \sqrt 2 \cos \left( {x + \dfrac{\pi }{4}} \right) = 0\left( {do\,k \in \mathbb{Z}} \right)\\
\Leftrightarrow x + \dfrac{\pi }{4} = \dfrac{\pi }{2} + n\pi \Leftrightarrow x = \dfrac{\pi }{4} + n\pi \left( {n \in \mathbb{Z}} \right)
\end{array}$
