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

$\begin{array}{rcl} & & \frac{\mathrm{TanA}}{1-\cot \mathrm{A}}+\frac{\cot \mathrm{A}}{1-\mathrm{TanA}}\\ & =& \frac{\mathrm{tanA} \mathrm{sinA}}{\sin \mathrm{A}-\cos \mathrm{A}}+\frac{\cot \mathrm{A} \mathrm{cosA}}{\cos \mathrm{A}-\sin \mathrm{A}}\\ & =& \frac{{\displaystyle \frac{{\sin }^2\mathrm{A}}{\cos \mathrm{A}}}-{\displaystyle \frac{{\cos }^2\mathrm{A}}{\sin \mathrm{A}}}}{\mathrm{Sin} \mathrm{A}-\mathrm{Cos} \mathrm{A}}\\ & =& \frac{{\sin }^3\mathrm{A}-{\cos }^3\mathrm{A}}{(\mathrm{sinA} \mathrm{cosA})(\mathrm{sinA}-\mathrm{cosA})}\\ & =& \frac{(\mathrm{sinA}-\mathrm{cosA})({\sin }^2\mathrm{A}+\mathrm{sinAcosA}+{\cos }^2\mathrm{A})}{\mathrm{sinA} \mathrm{cosA}(\mathrm{sinA}-\mathrm{cosA})}\\ & =& \sec \mathrm{A} \mathrm{cosecA}+1\end{array}$