$\sqrt{3} \operatorname{cosec} 20^{\circ} - \sec 20^{\circ}$ is equal to

The number of $x \in [0, 2\pi]$ for which $|\sqrt{2 \sin^4 x + 18 \cos^2 x} - \sqrt{2 \cos^4 x + 18 \sin^2 x}| = 1$ is

If $a = \sin \frac{\pi}{18} \sin \frac{5\pi}{18} \sin \frac{7\pi}{18}$ and $x$ is the solution of the equations $y = 2[x] + 2$ and $y = 3[x - 2]$,where $[x]$ denotes the greatest integer function of $x$,then $a$ is equal to:

If $\sin 2\theta + \sin 2\phi = \frac{1}{2}$ and $\cos 2\theta + \cos 2\phi = \frac{3}{2}$,then $\cos^2(\theta - \phi) =$

If $x \cos \theta = y \cos \left( \theta + \frac{2\pi}{3} \right) = z \cos \left( \theta + \frac{4\pi}{3} \right)$,then the value of $\frac{1}{x} + \frac{1}{y} + \frac{1}{z}$ is equal to

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