The value of the expression $ \left[\operatorname{cosec}\left(75^{\circ}+\theta\right)-\sec \left(15^{\circ}-\theta\right)-\tan \left(55^{\circ}+\theta\right)+\right. $ $ \left.\cot \left(35^{\circ}-\theta\right)\right] $ is
(A) $ -1 $
(B) 0
(C) 1
(D) $ \frac{3}{2} $

AcademicMathematicsNCERTClass 10

Given:

\( \left[\operatorname{cosec}\left(75^{\circ}+\theta\right)-\sec \left(15^{\circ}-\theta\right)-\tan \left(55^{\circ}+\theta\right)+\right. \) \( \left.\cot \left(35^{\circ}-\theta\right)\right] \)

To do:

We have to evaluate \( \left[\operatorname{cosec}\left(75^{\circ}+\theta\right)-\sec \left(15^{\circ}-\theta\right)-\tan \left(55^{\circ}+\theta\right)+\right. \) \( \left.\cot \left(35^{\circ}-\theta\right)\right] \).

Solution:  

We know that,

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

$cot\ (90^{\circ}- \theta) = tan\ \theta$

Therefore,

$\operatorname{cosec}\left(75^{\circ}+\theta\right)-\sec \left(15^{\circ}-\theta\right)-\tan \left(55^{\circ}+\theta\right)+\cot \left(35^{\circ}-\theta\right)$

$=\operatorname{cosec}(90^{\circ}-(15^{\circ}-\theta))-\sec \left(15^{\circ}-\theta\right)-\tan \left(55^{\circ}+\theta\right)+\cot (90^{\circ}-(55^{\circ}+\theta))$

$=\sec (15^{\circ}-\theta)-\sec \left(15^{\circ}-\theta\right)-\tan \left(55^{\circ}+\theta\right)+\tan (55^{\circ}+\theta)$

$=0$

raja
Updated on 10-Oct-2022 13:28:52

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