In a $\Delta ABC$,the value of $\sin A \cos B \cos C + \sin B \cos C \cos A + \sin C \cos A \cos B$ is:

The angle of elevation of the top of a vertical tower situated perpendicularly on a plane is observed as $60^{\circ}$ from a point $P$ on the same plane. From another point $Q$,$10\, m$ vertically above the point $P$,the angle of depression of the foot of the tower is $30^{\circ}$. The height of the tower is (in $m$)

If $x$ is real and $x+\frac{1}{x}=2 \cos \theta,$ then $\cos \theta=$

If the sum and difference of two angles are $\pi$ and $\frac{\pi}{12}$ respectively,then the values of the angles in degree measure are (in $^{\circ}$)

If $\frac{\cos \alpha}{\sin \beta} = n$ and $\frac{\cos \alpha}{\cos \beta} = m$,then the value of $\cos^2 \beta$ is:

$\Delta XYZ$ is right-angled at $Y$. If $m \angle X = 60^{\circ}$,then find the value of $\left(\sec Z + \frac{2}{\sqrt{3}}\right)$.