The value of $$\frac{{\sin {{23}^ \circ }\cos {{67}^ \circ } + \sec {{52}^ \circ }\sin {{38}^ \circ } + \cos {{23}^ \circ }\sin {{67}^ \circ } + {\text{cosec}}\,{{52}^ \circ }\cos {{38}^ \circ }}}{{{\text{cose}}{{\text{c}}^2}\,{{20}^ \circ } - {{\tan }^2}{{70}^ \circ }}}{\text{ is:}}$$

Solution (By Examveda Team)

$$\eqalign{ & \frac{{\sin {{23}^ \circ }\cos {{67}^ \circ } + \sec {{52}^ \circ }\sin {{38}^ \circ } + \cos {{23}^ \circ }\sin {{67}^ \circ } + {\text{cosec}}\,{{52}^ \circ }\cos {{38}^ \circ }}}{{{\text{cose}}{{\text{c}}^2}{{20}^ \circ } - {{\tan }^2}{{70}^ \circ }}} \cr & = \frac{{{{\sin }^2}{{23}^ \circ } + \frac{1}{{\cos {{52}^ \circ }}} \times \cos {{52}^ \circ } + {{\cos }^2}{{23}^ \circ } + \frac{1}{{\sin {{52}^ \circ }}} \times \sin {{52}^ \circ }}}{{{\text{cose}}{{\text{c}}^2}{{20}^ \circ } - {{\cot }^2}{{20}^ \circ }}} \cr & = \frac{{{{\sin }^2}{{23}^ \circ } + 1 + {{\cos }^2}{{23}^ \circ } + 1}}{1}\,\,\,\,\,\left[ {\therefore \,{{\sin }^2}\theta + {{\cos }^2}\theta = 1} \right] \cr & = 1 + 1 + 1 \cr & = 3 \cr} $$