If $\cos (A + B) = \alpha \cos A \cos B + \beta \sin A \sin B,$ then $(\alpha, \beta) =$

$\tan 100^\circ + \tan 125^\circ + \tan 100^\circ \tan 125^\circ = $

If $x \sin^{2} 60^{\circ} - \frac{3}{2} \sec 60^{\circ} \tan^{2} 30^{\circ} + \frac{4}{5} \sin^{2} 45^{\circ} \tan^{2} 60^{\circ} = 0$,then $x$ is

If $\cos A = \cos B \cos C$ and $A + B + C = \pi,$ then the value of $\cot B \cot C$ is

From two points,lying on the same horizontal line,the angles of elevation of the top of the pillar are $\theta$ and $\phi$ $(\theta < \phi)$. If the height of the pillar is $h$ $m$ and the two points lie on the same side of the pillar,then the distance between the two points is (in $m$):

If $\tan \theta = \frac{x \sin \phi}{1 - x \cos \phi}$ and $\tan \phi = \frac{y \sin \theta}{1 - y \cos \theta}$,then $\frac{x}{y} = $