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

The sum of the solutions in $x \in (0,4\pi )$ of the equation $4\sin \frac{x}{3}\left( {\sin \left( {\frac{{\pi  + x}}{3}} \right)} \right)\sin \left( {\frac{{2\pi  + x}}{3}} \right) = 1$ is

$cos (\alpha \,-\,\beta ) = 1$ and $cos (\alpha  +\beta ) = 1/e$ , where $\alpha , \beta \in [-\pi , \pi ]$ . Number of pairs of $(\alpha ,\beta )$ which satisfy both the equations is

The general solution of $sin\, x + sin \,5x = sin\, 2x + sin \,4x$ is :

The angles $\alpha, \beta, \gamma$ of a triangle satisfy the equations $2 \sin \alpha+3 \cos \beta=3 \sqrt{2}$ and $3 \sin \beta+2 \cos \alpha=1$. Then, angle $\gamma$ equals

Let $A=\left\{\theta \in R:\left(\frac{1}{3} \sin \theta+\frac{2}{3} \cos \theta\right)^2=\frac{1}{3} \sin ^2 \theta+\frac{2}{3} \cos ^2 \theta\right\}$.Then