Watching from the top of an $80 \, m$ high hill,the angle of elevation of the top of a tower is $30^\circ$ and the angle of depression of the base of the tower is $45^\circ$. Find the height of the tower in $m$.

$A$ pole stands on the ground. Watching from the top of the pole,the angle of depression of a stone on the ground $10 \, m$ away from the pole is $60^{\circ}$. Then the height of the pole is $\ldots \ldots \ldots \, m$. (Take $\sqrt{3} = 1.73$.)

$A$ window of a house is $h$ meters above the ground. From the window,the angles of elevation and depression of the top and the bottom of another house situated on the opposite side of the lane are found to be $\alpha$ and $\beta$,respectively. Prove that the height of the other house is $h(1 + \tan \alpha \cot \beta)$ meters.

Two trucks are parked in the same direction of a $300 \,m$ high tower. The angles of depression of these trucks from the top of the tower are found to be $45^{\circ}$ and $60^{\circ}$. Find the distance between these two trucks. (in $,m$)

Write 'True' or 'False' and justify your answer.
The value of $\tan \theta$ (where $\theta < 90^{\circ}$) increases as $\theta$ increases.

$A$ ladder leans against a wall. It makes an angle of $45^{\circ}$ with the ground and reaches a height of $10 \, m$ on the wall. Find the length of the ladder and the distance between the lower end of the ladder and the base of the wall.