The value of ( A1 - A + 1 - A A ) ÷ ( ^2A1 + cosec\,A + 1 ) is:

Answer: Option D

$$\begin{array}{rl}& \left(\frac{\sin A}{1-\cos A}+\frac{1-\cos A}{\sin A}\right)÷\left(\frac{{\cot }^{2}A}{1+\text{cosec}A}+1\right)\\ & =\left(\frac{{\sin }^{2}A+{\left(1+\cos A\right)}^2}{\sin A\left(1-\cos A\right)}\right)÷\left(\frac{\frac{{\cos }^{2}A}{{\sin }^{2}A}\times \sin A}{1+\sin A}+1\right)\\ & =\frac{{\sin }^{2}A+1+{\cos }^{2}A-2\cos A}{\sin A\left(1-\cos A\right)}÷\frac{{\cos }^{2}A+\sin A+{\sin }^{2}A}{\sin A\left(1+\sin A\right)}\\ & =\frac{2\left(1-\cos A\right)}{\sin A\left(1-\cos A\right)}\times \frac{\sin A\left(1+\sin A\right)}{\left(1+\sin A\right)}\\ & =2\end{array}$$