Prove that : \[\frac{\tan A}{1+\sec A}-\frac{\tan A}{1-\sec A}=2 \operatorname{cosec} A \]

\[\begin{aligned} & \text { LHS }=\frac{\tan A(1-\sec A)-\tan A(1+\sec A)}{1-\sec ^{2} A} \\ & =\frac{-2 \tan A \sec A}{-\tan ^{2} A} \\ & =2 \times \frac{\cos A}{\sin A} \times \frac{1}{\cos A} \\ & =2 \operatorname{cosec} A=R H S \end{aligned} \]