If $\sum_{r=1}^{50} \tan ^{-1} \frac{1}{2 r^{2}}=p$,then the value of $\tan p$ is :

The number of all possible values of $\theta$,where $0 < \theta < \pi$,for which the system of equations
$(y+z) \cos 3\theta = (xyz) \sin 3\theta$
$x \sin 3\theta = \frac{2 \cos 3\theta}{y} + \frac{2 \sin 3\theta}{z}$
$(xyz) \sin 3\theta = (y+2z) \cos 3\theta + y \sin 3\theta$
have a solution $(x_0, y_0, z_0)$ with $y_0 z_0 \neq 0$,is

If $\cot \theta + \tan \theta = m$ and $\sec \theta - \cos \theta = n$,then which of the following is correct?

Let $f_k(x) = \frac{1}{k}(\sin^k x + \cos^k x)$ where $x \in R$ and $k \ge 1$. Then $f_4(x) - f_6(x)$ is equal to:

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

Let $S = \{x \in R : \cos(x) + \cos(\sqrt{2}x) < 2\}$,then