If \ ( - 1 + 1 ) \^n = cosec\, - , then n = ?

Answer: Option B

$$\begin{array}{rl}& {\left\{\left(\frac{\sec \theta -1}{\sec \theta +1}\right)\right\}}^n=\text{cosec}\theta -\cot \theta \\ & \text{Put }\theta =45^{\circ }\\ & \text{So},{\left\{\left(\frac{\sec {45}^{\circ }-1}{\sec {45}^{\circ }+1}\right)\right\}}^n=\text{cosec}45^{\circ }-\cot 45^{\circ }\\ & \Rightarrow {\left\{\left(\frac{\sqrt{2}-1}{\sqrt{2}+1}\right)\right\}}^n=\sqrt{2}-1\\ & \text{Rationalize internally on left side, we get}\\ & \Rightarrow {\left\{{\left(\sqrt{2}-1\right)}^2\right\}}^n=\sqrt{2}-1\\ & \text{To equate }n\text{ should be }\frac{1}{2}\\ & \text{So, }n=0.5\end{array}$$