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

If $\cos x = \tan y$,$\cot y = \tan z$ and $\cot z = \tan x$,then $\sin x$ equals 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 $....$

If $\cos x+\cos y=p$ and $\sin x+\sin y=q$,then $\cos \left(\frac{x-y}{2}\right) = $

If $\tan \theta + \sec \theta = e^x$,then $\cos \theta$ equals

Let $f(\theta) = \sin \theta (\sin \theta + \sin 3\theta)$,then $f(\theta)$