If the solution of the equation $\log _{\cos x} \cot x+4 \log _{\sin x} \tan x=1, x \in\left(0, \frac{\pi}{2}\right), \quad$ is $\sin ^{-1}\left(\frac{\alpha+\sqrt{\beta}}{2}\right)$, where $\alpha, \beta$ are integers, then $\alpha+\beta$ is equal to:

The number of points in $(-\infty, \infty)$, for which $x^2-x \sin x-\cos x=0$, is

The real roots of the equation $cos^7x\,  +\,  sin^4x\,  =\,  1$  in the interval $(-\pi, \pi)$ are

Number of solution $(s)$ of equation $cosec\, \theta -cot \,\theta = 1$ in $[0,2 \pi]$ is-

The set of values of $x$ satisfying the equation,${2^{\tan \,\,\left( {x\,\, - \,\,{\textstyle{\pi  \over 4}}} \right)}}$ $- 2$${\left( {0.25} \right)^{\frac{{{{\sin }^2}\,\left( {x\,\, - \,\,{\textstyle{\pi  \over 4}}} \right)}}{{\cos \,\,2x}}}}$ $+ 1 = 0$, is :

If $\cos \theta + \cos 2\theta + \cos 3\theta = 0$, then the general value of $\theta $ is