# Solve this

Question:

$\left\{\frac{\cos 70^{\circ}}{\sin 20^{\circ}}+\frac{\cos 55^{\circ} \operatorname{cosec} 35^{\circ}}{\tan 5^{\circ} \tan 25^{\circ} \tan 45^{\circ} \tan 65^{\circ} \tan 85^{\circ}}\right\}$

Solution:

$\left\{\frac{\cos 70^{\circ}}{\sin 20^{\circ}}+\frac{\cos 55^{\circ} \operatorname{cosec} 35^{\circ}}{\tan 5^{\circ} \tan 25^{\circ} \tan 45^{\circ} \tan 65^{\circ} \tan 85^{\circ}}\right\}$

$=\left\{\frac{\cos \left(90^{\circ}-20^{\circ}\right)}{\sin 20^{\circ}}+\frac{\cos \left(90^{\circ}-35^{\circ}\right) \operatorname{cosec} 35^{\circ}}{\tan 5^{\circ} \tan 25^{\circ} \tan 45^{\circ} \tan 65^{\circ} \tan 85^{\circ}}\right\}$

$=\left\{\frac{\sin 20^{\circ}}{\sin 20^{\circ}}+\frac{\sin 35^{\circ} \operatorname{cosec} 35^{\circ}}{\tan 5^{\circ} \tan 25^{\circ} \tan 45^{\circ} \tan 65^{\circ} \tan 85^{\circ}}\right\} \quad\left(\because \cos \left(90^{\circ}-\theta\right)=\sin \theta\right)$

$=\left\{1+\frac{\sin 35^{*} \frac{1}{\sin 35^{\circ}}}{\tan 5^{\circ} \tan 25^{\circ} \tan 45^{\circ} \tan 65^{\circ} \tan 85^{\circ}}\right\} \quad\left(\because \operatorname{cosec} \theta=\frac{1}{\sin \theta}\right)$

$=\left\{1+\frac{1}{\tan 5^{\circ} \tan 25^{\circ} \tan 45^{\circ} \tan 65^{\circ} \tan 85^{\circ}}\right\}$

$=\left\{1+\frac{1}{\tan 5^{\circ} \tan 25^{\circ} \tan 45^{\circ} \tan \left(90^{\circ}-25^{\circ}\right) \tan \left(90^{\circ}-5^{\circ}\right)}\right\}$

$=\left\{1+\frac{1}{\tan 5^{\circ} \tan 25^{\circ} \tan 45^{\circ} \cot 25^{\circ} \cot 5^{\circ}}\right\} \quad\left(\because \tan \left(90^{\circ}-\theta\right)=\cot \theta\right)$

$=\left\{1+\frac{1}{\tan 5^{\circ} \tan 25^{\circ} \tan 45^{\circ} \frac{1}{\tan 25^{\circ}} \frac{1}{\tan 5^{\circ}}}\right\}$     $\left(\because \cot \theta=\frac{1}{\tan \theta}\right)$

$=\left\{1+\frac{1}{\tan 45^{\circ}}\right\}$

$=\left\{1+\frac{1}{1}\right\} \quad\left(\because \tan 45^{\circ}=1\right)$

$=1+1$

$=2$

Hence, $\left\{\frac{\cos 70^{\circ}}{\sin 20^{\circ}}+\frac{\cos 55^{\circ} \operatorname{cosec} 35^{\circ}}{\tan 5^{\circ} \tan 25^{\circ} \tan 45^{\circ} \tan 65^{\circ} \tan 85^{\circ}}\right\}=2$

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