$\left(\frac{{\displaystyle \sin 2A}}{{\displaystyle \sec A-1}}\right)\left(\frac{{\displaystyle \sec 2A}}{{\displaystyle \sec 2A+1}}\right)=$

$\begin{array}{l}G.E=\left(\frac{2\sin A\cos A}{1/\cos A-1}\right)\left(\frac{1/\cos 2A}{1/\cos 2A+1}\right)=\frac{2\sin A{\cos }^{2}A}{1-\cos A}\frac{1}{2{\cos }^{2}A}\\ =\frac{2\sin A}{1-\cos A}=\frac{2\sin A/2\cos A/2}{2{\sin }^{2}A/2}=\cot \frac{A}{2}.\end{array}$