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Answer:
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SoluTion :-
Expression
[tex]\sf \frac{cos\,58^{\circ}}{sin\,32^{\circ}}+\frac{sin\,22^{\circ}}{cos\,68^{\circ}}-\frac{cos\,38^{\circ}\:\:cosec\,52^{\circ}}{tab\,18^{\circ}\:\:tan\,25^{\circ}\:tan\,60^{\circ}\:tan\,72^{\circ}\:tan\,65^{\circ}}}[/tex]
Using Complimentary angles to simplify,
[tex]\sf {sin\,x=cos(90-x)\:\:,\:\:cos\,x=sin(90-x)\:\: and\:\:tan\,x=cot(90-x)}[/tex]
Now,
[tex]\sf {\frac{cos\,58^{\circ}}{cos\,(90-32)^{\circ}}+\frac{sin\,22^{\circ}}{sin\,(90-68)^{\circ}}-\frac{cos\,38^{\circ}\times\frac{1}{sin\,52^{\circ}}}{tab\,18^{\circ}\:\:tan\,25^{\circ}\:tan\,60^{\circ}\:cot\,(90-72)^{\circ}\:cot\,(90-65)^{\circ}}}[/tex]
[tex]\sf {\Rightarrow \frac{cos\,58^{\circ}}{cos\,58^{\circ}}+\frac{sin\,22^{\circ}}{sin\,22^{\circ}}-\frac{cos\,38^{\circ}\times\frac{1}{cos\,(90-52)^{\circ}}}{tab\,18^{\circ}\:\:tan\,25^{\circ}\:tan\,60^{\circ}\:cot\,18^{\circ}\:cot\,25^{\circ}}}[/tex]
[tex]\sf {\Rightarrow 1+1-\frac{cos\,38^{\circ}\times\frac{1}{cos\,38^{\circ}}}{tab\,18^{\circ}\:\:tan\,25^{\circ}\:tan\,60^{\circ}\:\times\frac{1}{tan\,18^{\circ}}\times\frac{1}{tan\,25^{\circ}}}}[/tex]
[tex]\sf {\Rightarrow 1+1-\frac{1}{1\times\:tan\,60^{\circ}\:\times1}}}[/tex]
[tex]\sf {\Rightarrow 1+1-\frac{1}{tan\,60^{\circ}}}}[/tex]
[tex]\sf {\Rightarrow 2-\frac{1}{\sqrt{3}}\ ( \because tan\,60^{\circ}=\sqrt{3} )}[/tex]
Value of Given Expression :-
[tex]\sf {\implies \frac{2\sqrt{3}-1}{\sqrt{3}}}[/tex]