If $(1 + \sin A)(1 + \sin B)(1 + \sin C) = (1 - \sin A)(1 - \sin B)(1 - \sin C)$,then each side is equal to

The product $(1+\tan 1^{\circ})(1+\tan 2^{\circ})(1+\tan 3^{\circ}) \dots (1+\tan 45^{\circ})$ equals

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

If $x=\log _e\left[\cot \left(\frac{\pi}{4}+\theta\right)\right]$ and $\theta \in\left(\frac{-\pi}{4}, \frac{\pi}{4}\right)$,then consider the following statements:
$I$. $\cosh x=\sec 2 \theta$
$II$. $\sinh x=-\tan 2 \theta$

If angle $\theta$ is divided into two parts such that the tangent of one part is $k$ times the tangent of the other and $\phi$ is their difference,then $\sin \theta = $

$\frac{2\sin \theta \tan \theta (1 - \tan \theta ) + 2\sin \theta \sec^2 \theta}{(1 + \tan \theta)^2} = $