Number of solution $(s)$ of the equation ${\cos ^2}2x + {\cos ^2}\frac{{5x}}{4} = \cos 2x\,{\cos ^2}5x$ in $\left[ {0,\frac{\pi }{3}} \right]$ is

Let $f:[0,2] \rightarrow R$ be the function defined by

$f ( x )=(3-\sin (2 \pi x )) \sin \left(\pi x -\frac{\pi}{4}\right)-\sin \left(3 \pi x +\frac{\pi}{4}\right)$

If $\alpha, \beta \in[0,2]$ are such that $\{x \in[0,2]: f(x) \geq 0\}=[\alpha, \beta]$, then the value of $\beta-\alpha$ is. . . . . . . . . 

If $\sec 4\theta - \sec 2\theta = 2$, then the general value of $\theta $ is

If $\theta $ and $\phi $ are acute satisfying $\sin \theta = \frac{1}{2},$ $\cos \phi = \frac{1}{3},$ then $\theta + \phi \in $

The number of integral values of $k$, for which the equation $7\cos x + 5\sin x = 2k + 1$ has a solution, is

The most general value of $\theta $ satisfying the equations $\sin \theta = \sin \alpha $ and $\cos \theta = \cos \alpha $ is