What is the value of $$\frac{{\cos {{40}^ \circ } - \cos {{140}^ \circ }}}{{\sin {{80}^ \circ } + \sin {{20}^ \circ }}}?$$

Solution(By Examveda Team)

$$\eqalign{ & \frac{{\cos {{40}^ \circ } - \cos {{140}^ \circ }}}{{\sin {{80}^ \circ } + \sin {{20}^ \circ }}} \cr & = \frac{{ - 2\sin \left( {\frac{{{{40}^ \circ } + {{140}^ \circ }}}{2}} \right)\sin \left( {\frac{{{{40}^ \circ } - {{140}^ \circ }}}{2}} \right)}}{{2\sin \left( {\frac{{{{80}^ \circ } + {{20}^ \circ }}}{2}} \right)\cos \left( {\frac{{{{80}^ \circ } - {{20}^ \circ }}}{2}} \right)}} \cr & = \frac{{ - 2\sin \left( {\frac{{{{180}^ \circ }}}{2}} \right)\sin \left( { - \frac{{{{100}^ \circ }}}{2}} \right)}}{{2\sin \left( {\frac{{{{100}^ \circ }}}{2}} \right)\cos \left( {\frac{{{{60}^ \circ }}}{2}} \right)}} \cr & = \frac{{ - 2\sin {{90}^ \circ } \times \sin \left( { - {{50}^ \circ }} \right)}}{{2\sin {{50}^ \circ } \times \cos {{30}^ \circ }}} \cr & = \frac{{ - 2\sin {{90}^ \circ } \times \left( { - \sin {{50}^ \circ }} \right)}}{{2\sin {{50}^ \circ } \times \cos {{30}^ \circ }}} \cr & = \frac{{2\sin {{90}^ \circ } \times \sin {{50}^ \circ }}}{{2\sin {{50}^ \circ } \times \cos {{30}^ \circ }}} \cr & = \frac{{\sin {{90}^ \circ }}}{{\cos {{30}^ \circ }}} \cr & = \frac{1}{{\frac{{\sqrt 3 }}{2}}} \cr & = \frac{2}{{\sqrt 3 }} \cr} $$