The value of A A + cosec\,A - A A - cosec\,A - 1 is:

Answer: Option B

$$\begin{array}{rl}& \frac{\sin A}{\cot A+\text{cosec}A}-\frac{\sin A}{\cot A-\text{cosec}A}-1\\ & =\sin A\left[\frac{\cot A-\text{cosec}A-\cot A-\text{cosec}A}{{\cot }^{2}A-\text{cose}{\text{c}}^{2}A}\right]-1\\ & =\frac{-2\sin A.\text{cosec}A}{-1}+1\\ & =2+1\\ & =3\end{array}$$