The number of solutions of the pair of equations $$$\,2{\sin ^2}\theta - \cos 2\theta = 0$$$ $$$2co{s^2}\theta - 3\sin \theta = 0$$$

in the interval $$\left[ {0,2\pi } \right]$$

The values of $$\theta \in \left( {0,2\pi } \right)$$ for which $$2\,{\sin ^2}\theta - 5\,\sin \theta + 2 > 0,$$ are

Let $$\theta \in \left( {0,{\pi \over 4}} \right)$$ and $${t_1} = {\left( {\tan \theta } \right)^{\tan \theta }},\,\,\,\,{t_2} = \,\,{\left( {\tan \theta } \right)^{\cot \theta }}$$, $${t_3}\, = \,\,{\left( {\cot \theta } \right)^{\tan \theta }}$$ and $${t_4}\, = \,\,{\left( {\cot \theta } \right)^{\cot \theta }},$$then