Official solution
\[\sin 2\theta + \tan 2\theta > 0\]
\[ \Rightarrow \sin 2\theta + {{\sin 2\theta } \over {\cos 2\theta }} > 0\]
\[ \Rightarrow \sin 2\theta {{(\cos 2\theta + 1)} \over {\cos 2\theta }} > 0 \Rightarrow \tan 2\theta (2{\cos ^2}\theta ) > 0\]
Note : \[\cos 2\theta \ne 0\]
\[ \Rightarrow 1 - 2{\sin ^2}\theta \ne \theta \Rightarrow \sin \theta \ne \pm {1 \over {\sqrt 2 }}\]
Now, \[\tan 2\theta (1 + \cos 2\theta ) > 0\]
\[ \Rightarrow \tan 2\theta > 0\] (as \[\cos 2\theta + 1 > 0\])
\[ \Rightarrow 2\theta \in \left( {0,{\pi \over 2}} \right) \cup \left( {\pi ,{{3\pi } \over 2}} \right) \cup \left( {2\pi ,{{5\pi } \over 2}} \right) \cup \left( {3\pi ,{{7\pi } \over 2}} \right)\]
\[ \Rightarrow \theta \in \left( {0,{\pi \over 4}} \right) \cup \left( {{\pi \over 2},{{3\pi } \over 4}} \right) \cup \left( {\pi ,{{5\pi } \over 4}} \right) \cup \left( {{{3\pi } \over 2},{{7\pi } \over 4}} \right)\]