All possible values of $\theta$ $\in$ [0, 2$\pi$] for which sin 2$\theta$ + tan 2$\theta$ > 0 lie in :

$\sin 2\theta + \tan 2\theta &gt; 0$<br><br>$\Rightarrow \sin 2\theta + {{\sin 2\theta } \over {\cos 2\theta }} &gt; 0$<br><br>$$ \Rightarrow \sin 2\theta {{(\cos 2\theta + 1)} \over {\cos 2\theta }} &gt; 0 \Rightarrow \tan 2\theta (2{\cos ^2}\theta ) &gt; 0$$<br><br>Note : $\cos 2\theta \ne 0$<br><br>$$ \Rightarrow 1 - 2{\sin ^2}\theta \ne \theta \Rightarrow \sin \theta \ne \pm {1 \over {\sqrt 2 }}$$<br><br>Now, $\tan 2\theta (1 + \cos 2\theta ) &gt; 0$<br><br>$\Rightarrow \tan 2\theta &gt; 0$ (as $\cos 2\theta + 1 &gt; 0$)<br><br>$$ \Rightarrow 2\theta \in \left( {0,{\pi \over 2}} \right) \cup \left( {\pi ,{{3\pi } \over 2}} \right) \cup \left( {2\pi ,{{5\pi } \over 2}} \right) \cup \left( {3\pi ,{{7\pi } \over 2}} \right)$$<br><br>$$ \Rightarrow \theta \in \left( {0,{\pi \over 4}} \right) \cup \left( {{\pi \over 2},{{3\pi } \over 4}} \right) \cup \left( {\pi ,{{5\pi } \over 4}} \right) \cup \left( {{{3\pi } \over 2},{{7\pi } \over 4}} \right)$$