The expression of $$\frac{{\cot \theta + \operatorname{cosec} \theta - 1}}{{\cot \theta + \operatorname{cosec} \theta + 1}}$$    is equal to?

$$\eqalign{ & \frac{{\cot \theta + \operatorname{cosec} \theta - 1}}{{\cot \theta + \operatorname{cosec} \theta + 1}} \cr & {\text{Put }}\theta = {45^ \circ } \cr & = \frac{{1 + \sqrt 2 - 1}}{{1 + \sqrt 2 + 1}} \cr & = \frac{{\sqrt 2 }}{{2 + \sqrt 2 }} \cr & = \frac{{\sqrt 2 }}{{\sqrt 2 \left( {\sqrt 2 + 1} \right)}} \cr & = \frac{1}{{\sqrt 2 + 1}} \cr & = \sqrt 2 - 1 \cr & {\text{Now option B}} \cr & \frac{{1 - \cos \theta }}{{\sin \theta }} \cr & = \frac{{1 - \frac{1}{{\sqrt 2 }}}}{{\frac{1}{{\sqrt 2 }}}} \cr & = \sqrt 2 - 1{\text{ }}\left( {{\text{Satisfy}}} \right) \cr} $$