The number of values of $x$ in the interval $\left(\frac{\pi}{4}, \frac{7 \pi}{4}\right)$ for which $14 \operatorname{cosec}^{2} x-2 \sin ^{2} x=21$ $-4 \cos ^{2} x$ holds, is

If $\cos \,\alpha  + \cos \,\beta  = \frac{3}{2}$ and $\sin \,\alpha  + \sin \,\beta  = \frac{1}{2}$ and $\theta $ is the the arithmetic mean of $\alpha $ and $\beta $ , then $\sin \,2\theta  + \cos \,2\theta $ is equal to 

If $\sin 2\theta = \cos \theta ,\,\,0 < \theta < \pi $, then the possible values of $\theta $ are

If $0 \le x < 2\pi $ , then the number of real values of $x,$ which satisfy the equation  $\cos x + \cos 2x + \cos 3x + \cos 4x = 0$ is  . .  .

If $\cos {40^o} = x$ and $\cos \theta = 1 - 2{x^2}$, then the possible values of $\theta $ lying between ${0^o}$ and ${360^o}$is

The number of values of $\alpha $ in $[0, 2\pi]$ for which $2\,{\sin ^3}\,\alpha  - 7\,{\sin ^2}\,\alpha  + 7\,\sin \,\alpha  = 2$ , is