The value of $$\frac{{4{{\tan }^2}{{30}^ \circ } + {{\sin }^2}{{30}^ \circ }{{\cos }^2}{{45}^ \circ } + {{\sec }^2}{{48}^ \circ } - {{\cot }^2}{{42}^ \circ }}}{{\cos {{37}^ \circ }\sin {{53}^ \circ } + \sin {{37}^ \circ }\cos {{53}^ \circ } + \tan {{18}^ \circ }\tan {{72}^ \circ }}}\,{\text{is:}}$$

Solution(By Examveda Team)

$$\eqalign{ & \frac{{4{{\tan }^2}{{30}^ \circ } + {{\sin }^2}{{30}^ \circ }{{\cos }^2}{{45}^ \circ } + {{\sec }^2}{{48}^ \circ } - {{\cot }^2}{{42}^ \circ }}}{{\cos {{37}^ \circ }\sin {{53}^ \circ } + \sin {{37}^ \circ }\cos {{53}^ \circ } + \tan {{18}^ \circ }\tan {{72}^ \circ }}} \cr & = \frac{{4{{\left( {\frac{1}{{\sqrt 3 }}} \right)}^2} + {{\left( {\frac{1}{2}} \right)}^2}{{\left( {\frac{1}{{\sqrt 2 }}} \right)}^2} + {\text{cose}}{{\text{c}}^2}{{42}^ \circ } - {{\cot }^2}{{42}^ \circ }}}{{\cos {{37}^ \circ }\cos {{37}^ \circ } + \sin {{37}^ \circ }\sin {{37}^ \circ } + 1}} \cr & = \frac{{4 \times \frac{1}{3} + \frac{1}{4} \times \frac{1}{2} + 1}}{{{{\cos }^2}{{37}^ \circ } + {{\sin }^2}{{37}^ \circ } + 1}} \cr & = \frac{{\frac{4}{3} + \frac{1}{8} + 1}}{{1 + 1}} \cr & = \frac{{32 + 3 + 24}}{{24 \times 2}} \cr & = \frac{{59}}{{48}} \cr} $$