LHS $= \frac{\tan A}{1+\sec A} - \frac{\tan A}{1-\sec A} = \frac{\tan A(1-\sec A) - \tan A(1+\sec A)}{(1+\sec A)(1-\sec A)} = \frac{\tan A[(1-\sec A)-(1+\sec A)]}{1-\sec^2 A} = \frac{\tan A(-2\sec A)}{-\tan^2 A}$ (since $1-\sec^2A=-\tan^2A$) $= \frac{2\sec A}{\tan A} = \frac{2 \cdot \frac{1}{\cos A}}{\frac{\sin A}{\cos A}} = \frac{2}{\sin A} = 2\,\text{cosec}\,A$ = RHS (Proved)