The expression $$\frac{{{{\left( {1 - \sin \theta + \cos \theta } \right)}^2}\left( {1 - \cos \theta } \right){{\sec }^3}\theta \,{\text{cose}}{{\text{c}}^2}\theta }}{{\left( {\sec \theta - \tan \theta } \right)\left( {\tan \theta + \cot \theta } \right)}},$$        0° < θ < 90°, is equal to:

Solution(By Examveda Team)

$$\eqalign{ & \frac{{{{\left( {1 - \sin \theta + \cos \theta } \right)}^2}\left( {1 - \cos \theta } \right){{\sec }^3}\theta {\text{cose}}{{\text{c}}^2}\theta }}{{\left( {\sec \theta - \tan \theta } \right)\left( {\tan \theta + \cot \theta } \right)}} \cr & {\text{put }}\theta = {30^ \circ } \cr & = \frac{{{{\left( {1 - \sin {{30}^ \circ } + \cos {{30}^ \circ }} \right)}^2}\left( {1 - \cos {{30}^ \circ }} \right){{\sec }^3}{{30}^ \circ }{\text{cose}}{{\text{c}}^2}{{30}^ \circ }}}{{\left( {\sec {{30}^ \circ } - \tan {{30}^ \circ }} \right)\left( {\tan {{30}^ \circ } + \cot {{30}^ \circ }} \right)}} \cr & = \frac{{{{\left( {1 - \frac{1}{2} + \frac{{\sqrt 3 }}{2}} \right)}^2}\left( {1 - \frac{{\sqrt 3 }}{2}} \right){{\left( {\frac{2}{{\sqrt 3 }}} \right)}^3}{{\left( 2 \right)}^2}}}{{\left( {\frac{2}{{\sqrt 3 }} - \frac{1}{{\sqrt 3 }}} \right)\left( {\frac{1}{{\sqrt 3 }} + \sqrt 3 } \right)}} \cr & = \frac{{{{\left( {\frac{{1 + \sqrt 3 }}{2}} \right)}^2}\left( {\frac{{2 - \sqrt 3 }}{2}} \right){{\left( {\frac{2}{{\sqrt 3 }}} \right)}^3}{{\left( 2 \right)}^2}}}{{\left( {\frac{1}{{\sqrt 3 }}} \right)\left( {\frac{{1 + 3}}{{\sqrt 3 }}} \right)}} \cr & = \frac{{\frac{{4 + 2\sqrt 3 }}{4} \times \frac{{2 - \sqrt 3 }}{2} \times \frac{8}{{3\sqrt 3 }} \times 4}}{{\left( {\frac{1}{{\sqrt 3 }}} \right)\left( {\frac{4}{{\sqrt 3 }}} \right)}} \cr & = \frac{{\left[ {{2^2} - {{\left( {\sqrt 3 } \right)}^2}} \right] \times \frac{8}{{3\sqrt 3 }}}}{{\frac{4}{3}}} \cr & = \frac{{\frac{8}{{3\sqrt 3 }}}}{{\frac{4}{3}}} \cr & = \frac{2}{{\sqrt 3 }} \cr & {\text{Option A: }}2\tan {30^ \circ } = \frac{2}{{\sqrt 3 }} \cr & {\text{Option B: }}\cot {30^ \circ } = \sqrt 3 \cr & {\text{Option C: }}\sin {30^ \circ } = \frac{1}{2} \cr & {\text{Option D: }}2\cos {30^ \circ } = 2 \times \frac{{\sqrt 3 }}{2} = \sqrt 3 \cr & {\text{Only option A is answer}}{\text{.}} \cr} $$