Observe that,at any instant,the minute and hour hands of a clock make two angles between them whose sum is $360^{\circ}$. At $6:15$,the difference between these two angles is $....^{\circ}$.

If $0 < \theta < \frac{\pi}{2}$ and $\sin \theta \cos \theta = \frac{12}{25}$,then $\sin^4 \theta + \cos^4 \theta$ is equal to

Let $\alpha = \frac{1}{\sin 60^{\circ} \sin 61^{\circ}} + \frac{1}{\sin 62^{\circ} \sin 63^{\circ}} + \dots + \frac{1}{\sin 118^{\circ} \sin 119^{\circ}}$. Then the value of $\left(\frac{\operatorname{cosec} 1^{\circ}}{\alpha}\right)^2$ is $....$

The geometric mean of $\tan 1^{\circ}, \tan 2^{\circ}, \ldots, \tan 89^{\circ}$ is

If $\sinh x = -\frac{4}{3}$,then $\sinh 2x + \cosh 2x = $

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:
$I$ : $\cosh x=\sec 2 \theta$
$II$ : $\sinh x=-\tan 2 \theta$
Then which one of the following options is true?