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