In $\Delta ABC$,$m \angle B = 90^{\circ}$. If $m \angle C = \theta$,then $\tan \theta = \dots$

$A$ vertical tower stands on a horizontal plane and is surmounted by a vertical flag staff of height $h$. At a point on the plane,the angles of elevation of the bottom and the top of the flag staff are $\alpha$ and $\beta$,respectively. Prove that the height of the tower is $\left(\frac{h \tan \alpha}{\tan \beta-\tan \alpha}\right)$.

The angle of elevation of the top of the tower from a point $x \, m$ away from the tower is $30^{\circ}$. Then the height of the tower is $\ldots \ldots \ldots \, m$.

Write 'True' or 'False' and justify your answer.
The angle of elevation of the top of a tower is $30^{\circ}$. If the height of the tower is doubled,then the angle of elevation of its top will also be doubled.

From a point $A$ on the ground,the angle of elevation of the top of a tower is found to be $45^{\circ}$. If the distance between point $A$ and the base of the tower is $x$ and the height of the tower is $y$,then which of the following holds true?

Observing an object from the point of observation,if one gets the angle of depression,then the object under observation is.........