If $\sin \theta + \text{cosec} \theta = 2$,the value of $\sin^{10} \theta + \text{cosec}^{10} \theta$ is

If $\frac{\sqrt{2} \sin \alpha}{\sqrt{1+\cos 2 \alpha}}=\frac{1}{7}$ and $\sqrt{\frac{1-\cos 2 \beta}{2}}=\frac{1}{\sqrt{10}}$ where $\alpha, \beta \in (0, \frac{\pi}{2})$,then $\tan (\alpha+2 \beta)$ is equal to

If $0 \leq \theta \leq 2 \pi$,$0 \leq \alpha \leq 2 \pi$ and $\sec ^{2018} \theta + \operatorname{cosec}^{2018} \alpha = 2$,then the value of $\cos ^{2020} \theta + \sin ^{2022} \alpha =$

If $\sinh x = -\frac{1}{2}$,then $\tanh 2x = $

The value of $\cos \frac{\pi}{7} \cos \frac{2\pi}{7} \cos \frac{3\pi}{7}$ is

The sum $S = \sin \theta + \sin 2\theta + \dots + \sin n\theta$ equals