What is the value of $$\frac{{{{\left[ {1 - \tan \left( {{{90}^ \circ } - \theta } \right)} \right]}^2}}}{{\left[ {{{\cos }^2}\left( {{{90}^ \circ } - \theta } \right)} \right]}} - 1 = ?$$

Solution (By Examveda Team)

$$\eqalign{ & \frac{{{{\left[ {1 - \tan \left( {{{90}^ \circ } - \theta } \right)} \right]}^2}}}{{\left[ {{{\cos }^2}\left( {{{90}^ \circ } - \theta } \right)} \right]}} - 1 \cr & {\text{By putting }}\theta = {45^ \circ } \cr & \Rightarrow \frac{{{{\left[ {1 - \tan \left( {{{90}^ \circ } - {{45}^ \circ }} \right)} \right]}^2}}}{{\left[ {{{\cos }^2}\left( {{{90}^ \circ } - {{45}^ \circ }} \right)} \right]}} - 1 \cr & \Rightarrow \frac{{{{\left[ {1 - \tan {{45}^ \circ }} \right]}^2}}}{{{{\cos }^2}{{45}^ \circ }}} - 1 \cr & \Rightarrow 0 - 1 \cr & \Rightarrow - 1 \cr & {\text{By satisfying in option A}} \cr & \Rightarrow - \sin 2\theta \cr & \Rightarrow - \sin {90^ \circ } \cr & \Rightarrow - 1 \cr} $$