Evaluate:
$ \frac{\cos 58^{\circ}}{\sin 32^{\circ}}+\frac{\sin 22^{\circ}}{\cos 68^{\circ}}-\frac{\cos 38^{\circ} \operatorname{cosec} 52^{\circ}}{\tan 18^{\circ} \tan 35^{\circ} \tan 60^{\circ} \tan 72^{\circ} \tan 55^{\circ}} $

Given:

\( \frac{\cos 58^{\circ}}{\sin 32^{\circ}}+\frac{\sin 22^{\circ}}{\cos 68^{\circ}}-\frac{\cos 38^{\circ} \operatorname{cosec} 52^{\circ}}{\tan 18^{\circ} \tan 35^{\circ} \tan 60^{\circ} \tan 72^{\circ} \tan 55^{\circ}} \).

To do:

We have to evaluate \( \frac{\cos 58^{\circ}}{\sin 32^{\circ}}+\frac{\sin 22^{\circ}}{\cos 68^{\circ}}-\frac{\cos 38^{\circ} \operatorname{cosec} 52^{\circ}}{\tan 18^{\circ} \tan 35^{\circ} \tan 60^{\circ} \tan 72^{\circ} \tan 55^{\circ}} \). 

Solution:  

We know that,

$cos\ (90^{\circ}- \theta) = sin\ \theta$

$tan\ (90^{\circ}- \theta) = cot\ \theta$

$tan\ \theta \times \cot\ \theta=1$

$sin\ \theta \times \operatorname{cosec}\ \theta=1$

Therefore,

$\frac{\cos 58^{\circ}}{\sin 32^{\circ}}+\frac{\sin 22^{\circ}}{\cos 68^{\circ}}-\frac{\cos 38^{\circ} \operatorname{cosec} 52^{\circ}}{\tan 18^{\circ} \tan 35^{\circ} \tan 60^{\circ} \tan 72^{\circ} \tan 55^{\circ}}=\frac{\cos( 90^{\circ}-32^{\circ})}{\sin 32^{\circ}} +\frac{\sin 22^{\circ}}{\cos( 90^{\circ}-22^{\circ})} -\frac{\cos( 90^{\circ}-22^{\circ}) \operatorname{cosec} 52^{\circ}}{\tan 18^{\circ}\tan 35^{\circ}\left(\sqrt{3}\right)\tan( 90^{\circ}-35^{\circ})\tan( 90^{\circ}-18^{\circ})}$

$=\frac{\sin 32^{\circ}}{\sin 32^{\circ}} +\frac{\sin 22^{\circ}}{\sin 22^{\circ}} -\frac{\sin 22^{\circ}\operatorname{cosec} 52^{\circ}}{\tan 18^{\circ}\tan 35^{\circ}\left(\sqrt{3}\right)\cot 35^{\circ}\cot 18^{\circ}}$

$=1+1-\frac{1}{( 1)\left(\sqrt{3}\right)( 1)}$

$=2-\frac{1}{\sqrt{3}}$

$=\frac{2\sqrt{3} -1}{\sqrt{3}}$

$=\frac{\left( 2\sqrt{3} -1\right)\sqrt{3}}{\sqrt{3} \times \sqrt{3}}$

$=\frac{2( 3) -1\left(\sqrt{3}\right)}{3}$

$=\frac{6-\sqrt{3}}{3}$

Hence, $\frac{\cos 58^{\circ}}{\sin 32^{\circ}}+\frac{\sin 22^{\circ}}{\cos 68^{\circ}}-\frac{\cos 38^{\circ} \operatorname{cosec} 52^{\circ}}{\tan 18^{\circ} \tan 35^{\circ} \tan 60^{\circ} \tan 72^{\circ} \tan 55^{\circ}}=\frac{6-\sqrt{3}}{3}$. 

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Updated on: 10-Oct-2022

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