Prove the following identities :

$\dfrac{\text{cosec A}}{\text{cosec A - 1}} + \dfrac{\text{cosec A}}{\text{cosec A + 1}}$ = 2 sec² A

Solving L.H.S. of the equation :

$\Rightarrow \dfrac{\text{cosec A}}{\text{cosec A - 1}} + \dfrac{\text{cosec A}}{\text{cosec A + 1}} \\[1em] \Rightarrow \dfrac{\text{cosec A(cosec A + 1) + cosec A(cosec A - 1)}}{\text{(cosec A - 1)(cosec A + 1)}} \\[1em] \Rightarrow \dfrac{\text{cosec}^2 A + \text{cosec A + cosec}^2 A - \text{cosec A}}{\text{cosec}^2 A - 1} \\[1em] \Rightarrow \dfrac{\text{2 cosec}^2 A}{\text{cot}^2 A} \\[1em] \Rightarrow \dfrac{2 \times \dfrac{1}{\text{sin}^2 A}}{\dfrac{\text{cos}^2 A}{\text{sin}^2 A}} \\[1em] \Rightarrow \dfrac{2 \times \dfrac{1}{\text{sin}^2 A} \times \text{sin}^2 A}{\text{cos}^2 A} \\[1em] \Rightarrow \dfrac{2}{\text{cos}^2 A} \\[1em] \Rightarrow 2\text{sec}^2 A.$

Since, L.H.S. = R.H.S.

Hence, proved that $\dfrac{\text{cosec A}}{\text{cosec A - 1}} + \dfrac{\text{cosec A}}{\text{cosec A + 1}}$ = 2 sec² A.