Express each one of the following in terms of trigonometric ratios of angles lying between $ 0^{\circ} $ and $ 45^{\circ} $$ \operatorname{cosec} 54^{\circ}+\sin 72^{\circ} $

Given:

\( \operatorname{cosec} 54^{\circ}+\sin 72^{\circ} \)

To do:

We have to express \( \operatorname{cosec} 54^{\circ}+\sin 72^{\circ} \) in terms of trigonometric ratios of angles lying between \( 0^{\circ} \) and \( 45^{\circ} \).

Solution:  

We know that,

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

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

Therefore,

$\operatorname{cosec} 54^{\circ}+\sin 72^{\circ}=\operatorname{cosec} (90^{\circ}-36^{\circ})+\sin (90^{\circ}-18^{\circ})$

$=\sec 36^{\circ}+\cos 18^{\circ}$

Therefore, $\operatorname{cosec} 54^{\circ}+\sin 72^{\circ}=sec 36^{\circ}+\cos 18^{\circ}$.    

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Updated on: 10-Oct-2022

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