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  • <img alt="\mathrm{Let\: S= \left \{ \theta \in \left [ -\pi,\pi \right ]-\left \{ \pm \frac{\pi}{2} \right \}:\sin \theta \tan \theta +\tan \theta = \sin 2\theta \right \}\cdot If\, T= \sum_{\theta

\mathrm{Let\: S= \left \{ \theta \in \left [ -\pi,\pi \right ]-\left \{ \pm \frac{\pi}{2} \right \}:\sin \theta \tan \theta +\tan \theta = \sin 2\theta \right \}\cdot If\, T= \sum_{\theta \, \in \, S}^{}\cos 2\theta,then\: T+n\left ( S \right ) \text{is equal to :}}

Option: 1

\mathrm{7+\sqrt{3}}


Option: 2

\mathrm{9}


Option: 3

\mathrm{8+\sqrt{3}}


Option: 4

\mathrm{10}


Answers (1)

best_answer

For S:
\mathrm{\tan \theta(\sin \theta+1)=\frac{2 \tan \theta}{1+\tan ^{2} \theta}}
\mathrm{\tan \theta=0,\quad \sin \theta+1=2 \cos ^{2} \theta}
                            \mathrm{\sin \theta=-1, \quad \sin \theta=\frac{1}{2}}
                                rejected
  \mathrm{\theta=0,-\pi, \pi}                             \mathrm{\theta=\frac{\pi}{6}, \frac{5 \pi}{6}}

\mathrm{\therefore T= \cos 0+\cos (-2 \pi)+\cos (2 \pi)+\cos \left(\frac{\pi}{3}\right)+\cos \left(\frac{5 \pi}{3}\right)}
       \mathrm{ T=3+\frac{1}{2}+\frac{1}{2}=4}
\mathrm{\therefore \quad T+n(S)=4+5=9}
The correct answer is option (B)

Posted by

mansi

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