The expression $\frac{\tan \mathrm{A}}{1-\cot \mathrm{A}}+\frac{\cot \mathrm{A}}{1-\tan \mathrm{A}}$ can be written as 

$\frac{\tan A}{1-cotA}+\frac{cotA}{1-\tan A}=\frac{{\sin }^{2}A}{\cos A\left(\sin A-\cos A\right)}+\frac{{\cos }^{2}A}{\sin A\left(\cos A-\sin A\right)}$

$=\frac{{\sin }^{3}A-{\cos }^{3}A}{\sin A\cos A\left(\sin A-\cos A\right)}=\frac{{\sin }^{2}A+{\cos }^{2}A+\sin A\cos A}{\sin A\cos A}$

$=1+\frac{1}{\sin A\cos A}=secA\cos ecA+1$