A) \[\frac{1}{2}\]
B) \[-\frac{1}{2}\]
C) \[\frac{3}{2}\]
D) \[\frac{-3}{2}\]
Correct Answer: B
Solution :
Given, \[\sqrt{3}\tan \theta =2\sin \theta \] \[\sqrt{3}\frac{\sin \theta }{\cos \theta }=2\sin \theta \] \[\therefore \,\,\,\,\,\,\,\,\,\cos \theta =\frac{\sqrt{3}}{2}\] \[\Rightarrow \,\cos \theta =\cos 30{}^\circ \Rightarrow \theta =30{}^\circ \] \[={{\sin }^{2}}\theta -{{\cos }^{2}}\theta \] \[={{\sin }^{2}}30{}^\circ -{{\cos }^{2}}30{}^\circ \] \[={{\left( \frac{1}{2} \right)}^{2}}-{{\left( \frac{\sqrt{3}}{2} \right)}^{2}}\] \[=\frac{1-3}{4}=\frac{2}{4}=-\frac{1}{2}\]
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