$x+y=\frac{\pi }{4}\Rightarrow \tan \left( x+y \right)=1$  $\cos \left( x-y \right)=\frac{3}{5}\Rightarrow \sin \left( x-y \right)=\pm \frac{4}{5}\Rightarrow \tan \left( x-y \right)=\pm \frac{4}{3}$  $2x=\left( x+y \right)+\left( x-y \right)\Rightarrow \tan 2x=\frac{\tan \left( x+y \right)+\tan \left( x-y \right)}{1-\tan \left( x+y \right)\tan \left( x-y \right)}$  $=\frac{1\pm \left( \frac{3}{4} \right)}{1-\left( 1 \right)\left( \pm \frac{4}{3} \right)}=\left\{ \begin{matrix}    \frac{1+\frac{4}{3}}{1-\frac{4}{3}}=-7  \\    \frac{1-\frac{4}{3}}{1+\frac{4}{3}}=-\frac{1}{7}  \\ \end{matrix} \right.\Rightarrow \cot 2x=\left\{ \begin{matrix}    -\frac{1}{7}  \\    -7  \\ \end{matrix} \right.$