Simplify $\frac{1}{\mathrm{cosec}A-\cot A}-\frac{1}{\mathrm{cosec}A+\cot A}$.

Since $(\mathrm{cosec}A-\cot A)(\mathrm{cosec}A+\cot A)={\mathrm{cosec}}^{2}A-{\cot }^{2}A=1$, each fraction equals the OTHER bracket. The expression becomes $(\mathrm{cosec}A+\cot A)-(\mathrm{cosec}A-\cot A)=2\cot A$. Answering $2\mathrm{cosec}A$ keeps the wrong pair of surviving terms; answering 0 assumes the two fractions are equal, which they are not.