Evaluate:
$ \frac{3 \cos 55^{\circ}}{7 \sin 35^{\circ}}-\frac{4\left(\cos 70^{\circ} \operatorname{cosec} 20^{\circ}\right)}{7\left(\tan 5^{\circ} \tan 25^{\circ} \tan 45^{\circ} \tan 65^{\circ} \tan 85^{\circ}\right)} $

Given:

\( \frac{3 \cos 55^{\circ}}{7 \sin 35^{\circ}}-\frac{4\left(\cos 70^{\circ} \operatorname{cosec} 20^{\circ}\right)}{7\left(\tan 5^{\circ} \tan 25^{\circ} \tan 45^{\circ} \tan 65^{\circ} \tan 85^{\circ}\right)} \).

To do:

We have to evaluate \( \frac{3 \cos 55^{\circ}}{7 \sin 35^{\circ}}-\frac{4\left(\cos 70^{\circ} \operatorname{cosec} 20^{\circ}\right)}{7\left(\tan 5^{\circ} \tan 25^{\circ} \tan 45^{\circ} \tan 65^{\circ} \tan 85^{\circ}\right)} \). 

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,

$\frac{3 \cos 55^{\circ}}{7 \sin 35^{\circ}}-\frac{4\left(\cos 70^{\circ} \operatorname{cosec} 20^{\circ}\right)}{7\left(\tan 5^{\circ} \tan 25^{\circ} \tan 45^{\circ} \tan 65^{\circ} \tan 85^{\circ}\right)}=\frac{3\cos( 90^{\circ}-35^{\circ})}{7\sin 35^{\circ}} -\frac{4[\cos( 90^{\circ}-20^{\circ}) \operatorname{cosec}20^{\circ}]}{7[\tan 5^{\circ}\tan 25^{\circ}( 1)\tan( 90^{\circ}-25^{\circ})\tan( 90^{\circ}-5^{\circ})]}$

$=\frac{3\sin 35^{\circ}}{7\sin 35^{\circ}} -\frac{4\sin 20^{\circ}\operatorname{cosec}20^{\circ}}{7\tan 5^{\circ}\tan 25^{\circ}\cot 25^{\circ}\cot 5^{\circ}}$

$=\frac{3}{7} -\frac{4}{7( 1)( 1)}$

$=\frac{3-4}{7}$

$=\frac{-1}{7}$

Hence, $\frac{3 \cos 55^{\circ}}{7 \sin 35^{\circ}}-\frac{4\left(\cos 70^{\circ} \operatorname{cosec} 20^{\circ}\right)}{7\left(\tan 5^{\circ} \tan 25^{\circ} \tan 45^{\circ} \tan 65^{\circ} \tan 85^{\circ}\right)}=\frac{-1}{7}$.

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Updated on: 10-Oct-2022

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