Official solution
Let f(x, y) = x + y - 1
\[ \because f\left( {1,2} \right).f\left( {\sin \theta ,\cos \theta } \right) > 0\]
\[ \Rightarrow 2\left[ {\sin \theta + \cos \theta - 1} \right] > 0\]
\[ \Rightarrow \sin \theta + \cos \theta > 1\]
\[ \Rightarrow \sin \left( {\theta + {\pi \over 4}} \right) > {1 \over {\sqrt 2 }}\]
\[ \Rightarrow \theta + {\pi \over 4} \in \left( {{\pi \over 4},{{3\pi } \over 4}} \right)\]
\[ \Rightarrow \theta \in \left( {0,{\pi \over 2}} \right)\]