If $${\cos ^3}x\,.\,\sin 2x = \sum\limits_{m = 1}^n {{a_m}\sin mx} $$ is identity in x, then

Total number of solutions of $$\left| {\cot x} \right| = \cot x + {1 \over {\sin x}},x \in [0,3\pi ]$$ is equal to

The equation $$(\cos \beta - 1){x^2} + (\cos \beta )x + \sin \beta = 0$$ in the variable x has real roots, then $$\beta$$ lies in the interval

The number of distinct solutions of the equation $${5 \over 4}{\cos ^2}2x + {\cos ^4}x + {\sin ^4}x + {\cos ^6}x = 2$$ in the interval [0, 2$$\pi$$] is