If $${\left\{ {\left( {\frac{{\sec \theta - 1}}{{\sec \theta + 1}}} \right)} \right\}^n} = {\text{cosec}}\,\theta - \cot \theta ,$$       then n = ?

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