Prove that : \[\frac{\tan A}{1-\cot A}+\frac{\cot A}{1-\tan A}=1+\sec A \operatorname{cosec} A \]

\[\begin{aligned} \mathrm{LHS} & =\frac{\dfrac{\sin A}{\cos A}}{\dfrac{(\sin A-\cos A)}{\sin A}}+\frac{\dfrac{\cos A}{\sin A}}{\dfrac{(\cos A-\sin A)}{\cos A}} \\ & =\frac{1}{(\sin A-\cos A)}\left[\frac{\sin ^{2} A}{\cos A}-\frac{\cos ^{2} A}{\sin A}\right] \\ & =\frac{1}{(\sin A-\cos A)} \times \frac{(\sin A-\cos A)\left(\sin ^{2} A+\cos ^{2} A+\sin A \cos A\right)}{\sin A \cos A} \\ & =\frac{1}{\sin A \cos A}+1 \\ & =1+\sec A \operatorname{cosec} A=\mathrm{RHS} \end{aligned} \]