The value of $$\frac{{\sin A}}{{\cot A + {\text{cosec}}\,A}} - \frac{{\sin A}}{{\cot A - {\text{cosec}}\,A}} - 1{\text{ is:}}$$

Solution (By Examveda Team)

$$\eqalign{ & \frac{{\sin A}}{{\cot A + {\text{cosec}}\,A}} - \frac{{\sin A}}{{\cot A - {\text{cosec}}\,A}} - 1 \cr & = \sin A\left[ {\frac{{\cot A - {\text{cosec}}\,A - \cot A - {\text{cosec}}\,A}}{{{{\cot }^2}A - {\text{cose}}{{\text{c}}^2}A}}} \right] - 1 \cr & = \frac{{ - 2\sin A.{\text{cosec}}\,A}}{{ - 1}} + 1 \cr & = 2 + 1 \cr & = 3 \cr} $$