What is the value of $$\frac{{1 + 2{{\cot }^2}\left( {{{90}^ \circ } - x} \right) - 2{\text{cosec}}\left( {{{90}^ \circ } - x} \right)\cot \left( {{{90}^ \circ } - x} \right)}}{{{\text{cosec}}\left( {{{90}^ \circ } - x} \right) - \cot \left( {{{90}^ \circ } - x} \right)}}?$$

Solution (By Examveda Team)

$$\eqalign{ & \frac{{1 + 2{{\cot }^2}\left( {{{90}^ \circ } - x} \right) - 2{\text{cosec}}\left( {{{90}^ \circ } - x} \right)\cot \left( {{{90}^ \circ } - x} \right)}}{{{\text{cosec}}\left( {{{90}^ \circ } - x} \right) - \cot \left( {{{90}^ \circ } - x} \right)}} \cr & \Rightarrow \frac{{1 + 2{{\tan }^2}x - 2\sec x\tan x}}{{\sec x - \tan x}} \cr & \Rightarrow \frac{{1 + 2\left( {{{\sec }^2}x - 1} \right) - 2\sec x\tan x}}{{\sec x - \tan x}} \cr & \Rightarrow \frac{{2{{\sec }^2}x - 1 - 2\sec x\tan x}}{{\sec x - \tan x}} \cr & \Rightarrow \frac{{2\sec x\left( {\sec x - \tan x} \right)}}{{\sec x - \tan x}} - \frac{1}{{\sec x - \tan x}} \cr & \Rightarrow 2\sec x - \frac{1}{{\sec x - \tan x}} \cr & \Rightarrow 2\sec x - \frac{1}{{\sec x - \tan x}} \times \frac{{\sec x + \tan x}}{{\sec x + \tan x}} \cr & \Rightarrow 2\sec x - \frac{{\sec x + \tan x}}{{{{\sec }^2}x - {{\tan }^2}x}} \cr & \Rightarrow 2\sec x - \sec x - \tan x \cr & \Rightarrow \sec x - \tan x \cr & \cr & {\bf{Alternative:}} \cr & \frac{{1 + 2{{\cot }^2}\left( {{{90}^ \circ } - x} \right) - 2{\text{cosec}}\left( {{{90}^ \circ } - x} \right)\cot \left( {{{90}^ \circ } - x} \right)}}{{{\text{cosec}}\left( {{{90}^ \circ } - x} \right) - \cot \left( {{{90}^ \circ } - x} \right)}} \cr & {\text{By putting }}x = {45^ \circ }{\text{ in equation}} \cr & \Rightarrow \frac{{1 + 2{{\tan }^2}{{45}^ \circ } - 2\sec {{45}^ \circ }\tan {{45}^ \circ }}}{{\sec {{45}^ \circ } - \tan {{45}^ \circ }}} \cr & \Rightarrow \frac{{1 + 2 - 2\sqrt 2 }}{{\sqrt 2 - 1}} \cr & \Rightarrow \frac{{3 - 2\sqrt 2 }}{{\sqrt 2 - 1}} \times \frac{{\left( {\sqrt 2 + 1} \right)}}{{\left( {\sqrt 2 + 1} \right)}} \cr & \Rightarrow 3\sqrt 2 - 2\sqrt 2 - 1 \cr & \Rightarrow \sqrt 2 \cr & {\text{By satisfying in options}} \cr & \Rightarrow \sec x - \tan x \Rightarrow \sqrt 2 - 1 \cr} $$