From an aeroplane vertically above a straight horizontal road, the angles of depression of two consecutive milestones on opposite sides of the aeroplane are observed to be $\alpha$ and $\beta .$ Show that the height (in miles) of aeroplane above the road is given by $\frac{\tan \alpha \tan \beta }{\tan \alpha +\tan \beta }$.

An observer, $1.5 \mathrm{m}$ tall, is $28.5 \mathrm{m}$ away from a tower $30 \mathrm{m}$ high. If the angle of elevation of the top of the tower from his eye is $p°$, then find the value of $p$.

A carpenter makes stools for electricians with a square top of side $0.5 \mathrm{m}$ and at a height of $1.5 \mathrm{m}$ above the ground. Also, each leg is inclined at an angle of ${60}^{\mathrm{o}}$ to the ground. If the length of the longer step (where steps are at equal distance.) is $q \mathrm{m}$, then find the value of $q$, correct up to three decimal places. [$\mathrm{Take} \sqrt{3}=1.732$]

From the top of a tower $\mathrm{h}$ metre high, the angles of depression of two objects, which are in the line with the foot of the tower are $\alpha$ and $\beta (\beta >\alpha )$. Find the value of $\frac{\mathrm{Distance} \mathrm{of} \mathrm{two} \mathrm{object}}{(\cot \alpha -\cot \beta )}$.