Let \( S = \left\{ \theta \in [-\pi, \pi] - \left\{ \pm \frac{\pi}{2} \right\} : \sin \theta \tan \theta + \tan \theta = \sin 2\theta \right\} \). If \( T = \ \displaystyle \sum_{\theta \in S} \cos 2\theta \), then \( T + n(S) \) is equal to:
\( 7 + \sqrt{3} \)
\( 9 \)
\( 8 + \sqrt{3} \)
\( 10 \)
