For an acute angle A it is given that $\mathrm{cosec}A-\cot A=\frac{1}{3}$.
Find $\mathrm{cosec}A$.

Because $(\mathrm{cosec}A-\cot A)(\mathrm{cosec}A+\cot A)={\mathrm{cosec}}^{2}A-{\cot }^{2}A=1$, the companion value is $\mathrm{cosec}A+\cot A=3$. Adding the two equations gives $2\mathrm{cosec}A=3+\frac{1}{3}=\frac{10}{3}$, so $\mathrm{cosec}A=\frac{5}{3}$ and $\cot A=\frac{4}{3}$. Answering $\frac{10}{3}$ forgets to halve; answering 3 reports the SUM instead of $\mathrm{cosec}A$ alone.