The value of $$\frac{{\left( {\cos {9^ \circ } + \sin {{81}^ \circ }} \right)\left( {\sec {9^ \circ } + {\text{cosec}}\,{{81}^ \circ }} \right)}}{{\sin {{56}^ \circ }\sec {{34}^ \circ } + \cos {{25}^ \circ }{\text{cosec}}\,{{65}^ \circ }}}{\text{ is:}}$$

Solution (By Examveda Team)

$$\eqalign{ & \frac{{\left( {\cos {9^ \circ } + \sin {{81}^ \circ }} \right)\left( {\sec {9^ \circ } + {\text{cosec}}\,{{81}^ \circ }} \right)}}{{\sin {{56}^ \circ }\sec {{34}^ \circ } + \cos {{25}^ \circ }{\text{cosec}}\,{{65}^ \circ }}} \cr & = \frac{{\left( {\cos {9^ \circ } + \cos {9^ \circ }} \right)\left( {\sec {9^ \circ } + \sec {9^ \circ }} \right)}}{{\cos {{34}^ \circ }\sec {{34}^ \circ } + \cos {{25}^ \circ }\sec {{25}^ \circ }}} \cr & = \frac{{2\cos {9^ \circ } \times 2\sec {9^ \circ }}}{{1 + 1}} \cr & = \frac{{4 \times \cos {9^ \circ }\sec {9^ \circ }}}{2} \cr & = 2 \cr} $$