If x = 2 1 + + , then the value of 1 - + 1 + is

Answer: Option B

$$\begin{array}{rl}& x=\frac{2\sin \theta }{1+\cos \theta +\sin \theta }\\ & \text{Let’s put the value of }\theta =45^{\circ }\\ & \text{So, }x=\frac{2\times \frac{1}{\sqrt{2}}}{1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}}\\ & \left\{\text{Because }\sin {45}^{\circ }=\frac{1}{\sqrt{2}}\text{ and}\cos {45}^{\circ }=\frac{1}{\sqrt{2}}\right\}\\ & x=\frac{\sqrt{2}}{\frac{\left(\sqrt{2}+2\right)}{\sqrt{2}}}\\ & x=\frac{\sqrt{2}}{\left(1+\sqrt{2}\right)}\\ & \text{Now, check in the required value}\\ & \frac{1-\cos \theta +\sin \theta }{1+\sin \theta }\\ & =\frac{1-\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}}{1+\frac{1}{\sqrt{2}}}\\ & =\frac{\sqrt{2}}{\left(\sqrt{2}+1\right)}\\ & \text{This values is same as }x.\\ & \text{Hence answer is }x.\end{array}$$