Prove that :

$\dfrac{\text{cot}^2 A}{\text{cosec A - 1}} - 1$ = cosec A

Solving L.H.S. of the equation :

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

By formula,

cot² A + 1 = cosec² A

$\Rightarrow \dfrac{\text{cosec}^2 A - \text{cosec A}}{\text{cosec A - 1}} \\[1em] \Rightarrow \dfrac{\text{cosec A(cosec A - 1)}}{\text{cosec A - 1}} \\[1em] \Rightarrow \text{cosec A}.$

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

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