If $|\cos x + \sin x| + |\cos x - \sin x| = 2 \sin x$ for $x \in [0, 2\pi]$,then the maximum integral value of $x$ is:

The geometric mean of $\tan 1^{\circ}, \tan 2^{\circ}, \ldots, \tan 89^{\circ}$ is

If $r \sin \theta = 3$ and $r = 4(1 + \sin \theta)$ for $0 \le \theta \le 2\pi$,then $\theta = $

Let $\tan \alpha, \tan \beta$ and $\tan \gamma$ (where $\alpha, \beta, \gamma \neq \frac{(2n-1)\pi}{2}, n \in N$) be the slopes of three line segments $OA, OB$ and $OC$ respectively,where $O$ is the origin. If the circumcentre of $\Delta ABC$ coincides with the origin and its orthocentre lies on the $y$-axis,then the value of $\left(\frac{\cos 3\alpha + \cos 3\beta + \cos 3\gamma}{\cos \alpha \cos \beta \cos \gamma}\right)^2$ is equal to:

The value of $\left(\sin 70^{\circ}\right)\left(\cot 10^{\circ} \cot 70^{\circ}-1\right)$ is

$16 \sin(20^{\circ}) \sin(40^{\circ}) \sin(80^{\circ})$ is equal to