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Express each one of the following in terms of trigonometric ratios of angles lying between $ 0^{\circ} $ and $ 45^{\circ} $$ \tan 65^{\circ}+\cot 49^{\circ} $
Given:
\( \tan 65^{\circ}+\cot 49^{\circ} \)
To do:
We have to express \( \tan 65^{\circ}+\cot 49^{\circ} \) in terms of trigonometric ratios of angles lying between \( 0^{\circ} \) and \( 45^{\circ} \).
Solution:
We know that,
$cot\ (90^{\circ}- \theta) = tan\ \theta$
$tan\ (90^{\circ}- \theta) = cot\ \theta$
Therefore,
$\tan 65^{\circ}+\cot 49^{\circ}=\tan (90^{\circ}-25^{\circ})+\cot (90^{\circ}-41^{\circ})$
$=\cot 25^{\circ}+\tan 41^{\circ}$
Therefore, $\tan 65^{\circ}+\cot 49^{\circ}=\cot 25^{\circ}+\tan 41^{\circ}$.
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