The expression of + cosec - 1 + cosec + 1 is equal to?

Answer: Option B

$$\begin{array}{rl}& \frac{\cot \theta +\mathrm{cosec}\theta -1}{\cot \theta +\mathrm{cosec}\theta +1}\\ & \text{Put }\theta =45^{\circ }\\ & =\frac{1+\sqrt{2}-1}{1+\sqrt{2}+1}\\ & =\frac{\sqrt{2}}{2+\sqrt{2}}\\ & =\frac{\sqrt{2}}{\sqrt{2}\left(\sqrt{2}+1\right)}\\ & =\frac{1}{\sqrt{2}+1}\\ & =\sqrt{2}-1\\ & \text{Now option B}\\ & \frac{1-\cos \theta }{\sin \theta }\\ & =\frac{1-\frac{1}{\sqrt{2}}}{\frac{1}{\sqrt{2}}}\\ & =\sqrt{2}-1\left(\text{Satisfy}\right)\end{array}$$