Evaluate the following:
$ \sec 50^{\circ} \sin 40^{\circ}+\cos 40^{\circ} \operatorname{cosec} 50^{\circ} $

Given:

\( \sec 50^{\circ} \sin 40^{\circ}+\cos 40^{\circ} \operatorname{cosec} 50^{\circ} \)

To do:

We have to evaluate \( \sec 50^{\circ} \sin 40^{\circ}+\cos 40^{\circ} \operatorname{cosec} 50^{\circ} \).

Solution:  

We know that,

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

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

$\sec\ \theta\ cos\ \theta=1$

Therefore,

$\sec 50^{\circ} \sin 40^{\circ}+\cos 40^{\circ} \operatorname{cosec} 50^{\circ}=\sec 50^{\circ} \sin (90^{\circ}-50^{\circ})+\cos 40^{\circ}{\operatorname{cosec} (90^{\circ}-40^{\circ})$

$=\sec 50^{\circ} \cos 50^{\circ}+\cos 40^{\circ} \sec 40^{\circ}$

$=1+1$

$=2$

Therefore, $\sec 50^{\circ} \sin 40^{\circ}+\cos 40^{\circ} \operatorname{cosec} 50^{\circ}=2$.  

Tutorialspoint

Simply Easy Learning

Updated on: 10-Oct-2022

40 Views

Kickstart Your Career

Get certified by completing the course

Get Started