L.H.S.$=\frac{\tan A}{1+\sec A}-\frac{\tan A}{1-\sec A}=\frac{\tan A(1-\sec A-1-\sec A)}{(1+\sec A)(1-\sec A)}$

$=\frac{\tan A(-2 \sec A)}{\left(1-\sec ^{2} A\right)}=\frac{2 \tan A \cdot \sec A}{\left(\sec ^{2} A-1\right)} \quad\left[\because  (a+b)(a-b)=a^{2}-b^{2}\right]$

$=\frac{2 \tan A \cdot \sec A}{\tan ^{2} A} \quad\left[\because \sec ^{2} A-\tan ^{2} A=1\right]\left[\because \sec \theta=\frac{1}{\cos \theta}\right.$ and $\left.\tan \theta=\frac{\sin \theta}{\cos \theta}\right]$

$=\frac{2 \sec A}{\tan A}=\frac{2}{\sin A}=2 \operatorname{cosec} A=$R.H.S. $\left[\because \operatorname{cosec} \theta=\frac{1}{\sin \theta}\right]$