$\dfrac{\operatorname{cosec} A}{\operatorname{cosec} A - 1} + \dfrac{\operatorname{cosec} A}{\operatorname{cosec} A + 1} = ?$

Over a common denominator the numerators add to $\dfrac{\operatorname{cosec} A(\operatorname{cosec} A+1) + \operatorname{cosec} A(\operatorname{cosec} A-1)}{\operatorname{cosec}^2 A - 1} = \dfrac{2\operatorname{cosec}^2 A}{\cot^2 A}$. Since $\operatorname{cosec}^2 A = 1+\cot^2 A$... evaluating gives $\dfrac{2\operatorname{cosec}^2 A}{\operatorname{cosec}^2 A - 1} = \dfrac{2/\sin^2 A}{\cos^2 A/\sin^2 A} = \dfrac{2}{\cos^2 A} = 2\sec^2 A$.