A) \[\frac{7}{22}\]
B) \[\frac{5}{22}\]
C) \[\frac{9}{22}\]
D) \[\frac{22}{5}\]
Correct Answer: B
Solution :
Given,\[\cos \theta =-\frac{\sqrt{3}}{2}<0\]and\[\theta \]does not lie in third quadrant. \[\therefore \,\,\theta \]must be lying in 2nd quadrant. \[\Rightarrow \] \[\tan \theta =-\frac{1}{\sqrt{3}}\] and \[\cot \theta =-\sqrt{3}\] ... (i) Also,\[\alpha \]lies in 3rd quadrant and\[\sin \alpha =-\frac{3}{5}\] \[\therefore \] \[\tan \alpha =\frac{3}{4}\]and\[\cos \alpha =-\frac{4}{5}\] ... (ii) \[\therefore \] \[\frac{2\tan \alpha +\sqrt{3}\tan \theta }{{{\cot }^{2}}\theta +\cos \alpha }=\frac{2\cdot \frac{3}{4}-\sqrt{3}\cdot \frac{1}{\sqrt{3}}}{3-\frac{4}{5}}\] \[=\frac{\frac{3}{2}-1}{3-\frac{4}{5}}=\frac{5}{22}\]
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