If $\sin \theta + \operatorname{cosec} \theta = 2$,then the value of $\sin^{10} \theta + \operatorname{cosec}^{10} \theta$ is equal to

Let $\alpha = 3 + 4 + 8 + 9 + 13 + 14 + \dots$ up to $40$ terms. If $(\tan \beta)^{1020}$ is a root of the equation $x^2 + x - 2 = 0$,where $\beta \in (0, \frac{\pi}{2})$,then $\sin^2 \beta + 3 \cos^2 \beta$ is equal to:

If $\sinh x = \frac{12}{5}$,then $\sinh 3x + \cosh 3x = $

Let $E = \left( 1 - \frac{\cos 61^\circ}{\cos 1^\circ} \right) \left( 1 - \frac{\cos 62^\circ}{\cos 2^\circ} \right) \dots \left( 1 - \frac{\cos 119^\circ}{\cos 59^\circ} \right)$,then $E$ is equal to:

If $\frac{1}{6} \sin \theta, \cos \theta$ and $\tan \theta$ are in geometric progression,then the solution set of $\theta$ is

If $\alpha_1, \alpha_2, \cdots, \alpha_n$ are in $A$.$P$. with common difference $\theta$,then the sum of the series $\sec \alpha_1 \sec \alpha_2 + \sec \alpha_2 \sec \alpha_3 + \cdots + \sec \alpha_{n-1} \sec \alpha_n = k(\tan \alpha_n - \tan \alpha_1)$,where $k=$