There is a $100\, m$ high tower at some distance from a cliff. The angle of elevation of the top of the cliff from the base of the tower is $45^{\circ}$ and the angle of depression of the base of the cliff from the top of the tower is $30^{\circ}$. Find the height of the cliff and also the distance between the tower and the cliff.

(N/A) Let $h$ be the height of the cliff and $d$ be the distance between the tower and the cliff.
From the base of the tower to the top of the cliff: $\tan(45^{\circ}) = \frac{h}{d} \implies 1 = \frac{h}{d} \implies h = d$.
From the top of the tower to the base of the cliff: $\tan(30^{\circ}) = \frac{100}{d} \implies \frac{1}{\sqrt{3}} = \frac{100}{d} \implies d = 100\sqrt{3} \approx 173.2\, m$.
Since $h = d$,the height of the cliff is $173.2\, m$ and the distance between them is $173.2\, m$.