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

$=\frac{{\tan }^{2}A}{\tan A-1}+\frac{\cot A}{1-\tan A}$

$=\frac{1}{\tan A-1}\left[{\tan }^{2}A-\frac{1}{\tan A}\right]$

$=\frac{1}{\tan A-1}\left[\frac{{\tan }^{3}A-1}{\tan A}\right]$

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

$=\tan A+\cot A+1$

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

$=\frac{{\sin }^{2}A+{\cos }^{2}A}{\cos A\sin A}+1$

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

$=\sec A\cos ecA+1$