If $${\text{0}} < {\text{A}} < {90^ \circ }{\text{,}}$$   then the value of $$\frac{1}{2}\cot {\text{A}}$$ $$\left[ {\frac{{1 + \left( {\operatorname{sec A} - {\text{tan A}}} \right)}}{{\operatorname{cosecA} \left( {\sec {\text{A}} - {\text{tan A}}} \right)}}} \right]$$     = ?

Solution(By Examveda Team)

$$\eqalign{ & {\text{According to the question,}} \cr & {\text{Put A}} = {45^ \circ } \cr & \Rightarrow \frac{1}{2} \times \cot {45^ \circ } \cr & \left[ {\frac{{1 + \left( {\sec {{45}^ \circ } + \tan {{45}^ \circ }} \right)}}{{{\text{cosec }}{{45}^ \circ }\left( {\sec {{45}^ \circ } - \tan {{45}^ \circ }} \right)}}} \right] \cr & \Rightarrow \frac{1}{2}\left[ {\frac{{1 + {{\left( {\sqrt 2 - 1} \right)}^2}}}{{\sqrt 2 \times \left( {\sqrt 2 - 1} \right)}}} \right] \cr & \Rightarrow \frac{1}{2}\left[ {\frac{{1 + 2 + 1 - 2\sqrt 2 }}{{2 - \sqrt 2 }}} \right] \cr & \Rightarrow \frac{1}{2}\left[ {\frac{{4 - 2\sqrt 2 }}{{2 - \sqrt 2 }}} \right] \cr & \Rightarrow \frac{1}{2} \times 2\left[ {\frac{{2 - \sqrt 2 }}{{2 - \sqrt 2 }}} \right] \cr & \Rightarrow 1 \cr} $$