# 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) $\frac{3}{2}$

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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,

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

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

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

$=0$

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