The value of $$\frac{{\left( {\cos {9^ \circ } + \sin {{81}^ \circ }} \right)\left( {\sec {9^ \circ } + {\text{cosec}}\,{\text{8}}{{\text{1}}^ \circ }} \right)}}{{{\text{cose}}{{\text{c}}^2}71 + {{\cos }^2}{{15}^ \circ } - {{\tan }^2}{{19}^ \circ } + {{\cos }^2}{{75}^ \circ }}}{\text{is:}}$$

Solution(By Examveda Team)

$$\eqalign{ & \frac{{\left( {\cos {9^ \circ } + \sin {{81}^ \circ }} \right)\left( {\sec {9^ \circ } + {\text{cosec}}\,{\text{8}}{{\text{1}}^ \circ }} \right)}}{{{\text{cose}}{{\text{c}}^2}71 + {{\cos }^2}{{15}^ \circ } - {{\tan }^2}{{19}^ \circ } + {{\cos }^2}{{75}^ \circ }}} \cr & = \frac{{\left( {\cos {9^ \circ } + \cos {9^ \circ }} \right)\left( {\sec {9^ \circ } + \sec {9^ \circ }} \right)}}{{{\text{cose}}{{\text{c}}^2}71 + {{\cos }^2}{{15}^ \circ } - {{\cot }^2}{{71}^ \circ } + {{\sin }^2}{{15}^ \circ }}} \cr & = \frac{{\left( {2\cos {9^ \circ }} \right)\left( {2\sec {9^ \circ }} \right)}}{{1 + 1}} \cr & = 2 \cr} $$