Evaluate the following:
$ \frac{\sec 70^{\circ}}{\operatorname{cosec} 20^{\circ}}+\frac{\sin 59^{\circ}}{\cos 31^{\circ}} $

Given:

\( \frac{\sec 70^{\circ}}{\operatorname{cosec} 20^{\circ}}+\frac{\sin 59^{\circ}}{\cos 31^{\circ}} \)

To do:

We have to evaluate \( \frac{\sec 70^{\circ}}{\operatorname{cosec} 20^{\circ}}+\frac{\sin 59^{\circ}}{\cos 31^{\circ}} \).

Solution:  

We know that,

$sec\ (90^{\circ}- \theta) = cosec\ \theta$

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

Therefore,

$\frac{\sec 70^{\circ}}{\operatorname{cosec} 20^{\circ}}+\frac{\sin 59^{\circ}}{\cos 31^{\circ}}=\frac{\sec (90^{\circ}-20^{\circ})}{\operatorname{cosec} 20^{\circ}}+\frac{\sin (90^{\circ}-31^{\circ})}{\cos 31^{\circ}}$

$=\frac{\operatorname{cosec} 20^{\circ}}{\operatorname{cosec} 20^{\circ}}+\frac{\cos 31^{\circ}}{\cos 31^{\circ}}$

$=1+1$

$=2$

Therefore, $\frac{\sec 70^{\circ}}{\operatorname{cosec} 20^{\circ}}+\frac{\sin 59^{\circ}}{\cos 31^{\circ}}=2$.   

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Simply Easy Learning

Updated on: 10-Oct-2022

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