If $\sec(x) = \cosh(\theta)$,then find $\tanh^2\left(\frac{\theta}{2}\right)$.

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

If $3 \sin^{2} x - 8 \sin x + 4 = 0$ and $x \in \left(\frac{\pi}{2}, \pi\right)$,then $\tan x = $

Let $\theta, \phi \in [0, 2\pi]$ be such that $2 \cos \theta(1-\sin \phi) = \sin^2 \theta \left(\tan \frac{\theta}{2} + \cot \frac{\theta}{2}\right) \cos \phi - 1$,$\tan (2\pi - \theta) > 0$ and $-1 < \sin \theta < -\frac{\sqrt{3}}{2}$. Then $\phi$ cannot satisfy

$4 \cos \frac{\pi}{7} \cos \frac{\pi}{5} \cos \frac{2 \pi}{7} \cos \frac{2 \pi}{5} \cos \frac{4 \pi}{7} = $

$\operatorname{sech}^2\left(\tanh ^{-1} \frac{1}{2}\right)+\operatorname{cosech}^2\left(\operatorname{coth}^{-1} 3\right)=$