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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} $
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,$\operatorname{cosec}\left(75^{\circ}+\theta\right)-\sec \left(15^{\circ}-\theta\right)-\tan \left(55^{\circ}+\theta\right)+\cot \left(35^{\circ}-\theta\right)$
$=\operatorname{cosec}(90^{\circ}-(15^{\circ}-\theta))-\sec \left(15^{\circ}-\theta\right)-\tan \left(55^{\circ}+\theta\right)+\cot (90^{\circ}-(55^{\circ}+\theta))$
$=\sec (15^{\circ}-\theta)-\sec \left(15^{\circ}-\theta\right)-\tan \left(55^{\circ}+\theta\right)+\tan (55^{\circ}+\theta)$
$=0$