Question

Find the value of : $\frac{\cos {38}^{\circ }\csc {52}^{\circ }}{\tan {18}^{\circ }\tan {35}^{\circ }\tan {60}^{\circ }\tan {72}^{\circ }\tan {55}^{\circ }}=$

• A. $\sqrt{3}$
• B. $\frac{1}{3}$
• C. $\frac{1}{\sqrt{3}}$
• D. $\frac{2}{\sqrt{3}}$

Answer 1

The correct option is C $\frac{1}{\sqrt{3}}$

Given:

$\frac{\cos {38}^o\csc {52}^o}{\tan {18}^o\tan {35}^o\tan {60}^o\tan {72}^o\tan {55}^o}$

$=\frac{\cos {38}^o\csc \left({90}^o-{38}^o\right)}{\left(\tan {18}^o\times \tan {72}^o\right)\times \left(\tan {35}^o\times \tan {55}^o\right)\times \tan {60}^o}$

$=\frac{\cos {38}^o\sec {38}^o}{\tan {18}^o\tan \left({90}^o-{18}^o\right)\tan {35}^o\tan \left({90}^o-{35}^o\right)\tan {60}^o}$

( we know that $\tan ({90}^o-\theta )=\cot \theta$ and $\cot \theta =\frac{1}{\tan \theta }$and $\sec \theta =\frac{1}{\cos \theta }$ and $\tan {60}^o=\sqrt{3})$

$\therefore \frac{\cos {38}^o\csc {52}^o}{\tan {18}^o\tan {35}^o\tan {60}^o\tan {72}^o\tan {55}^o}$

$=\frac{1}{\left(\tan {18}^o\cot {18}^o\right)\left(\tan {35}^o\cot {35}^o\right)\sqrt{3}}$

$=\frac{1}{\sqrt{3}}$