If $\cot 3A = x,$ then the value of $x$ is

$A$ man observes the angle of elevation of the top of a tower having a height of $100 \, m$ as $30^{\circ}$. After walking a certain distance towards the tower,he observes the angle of elevation as $45^{\circ}$. What is the distance through which the man walked towards the tower? (in $m$)

If $a \tan \theta = b$,then $a \cos 2\theta + b \sin 2\theta = $

The value of $\tan 9^{\circ} - \tan 27^{\circ} - \tan 63^{\circ} + \tan 81^{\circ}$ is

For all $\theta,$ the value of $\sqrt{\frac{1+\sin \theta}{1-\sin \theta}}$ is:

If $\sin (\theta + \alpha ) = a$ and $\sin (\theta + \beta ) = b,$ then $\cos 2(\alpha - \beta ) - 4ab\cos (\alpha - \beta )$ is equal to