<s>[INST] Depression of a car 
A person on the summit of mountain observes that the angles of depression of a car moving on a straight road at three consecutive mile stones are x, y and z respectively. Prove that the height of the mountain is $\sqrt{\frac{2}{\cot^2x-2\cot^2y+\cot^2z}}$

I am getting the height as $\frac{1}{\cot y-\cot z}$ or as $\frac{1}{\cot x - \cot y}$. Not getting the desired expression.
 [/INST] 
Let the distance between the foot of the mountain and the first mile stone be $a$ . Then the distances between the foot of the mountain and the consecutive milestones are $a+1$ and $a+2$.
Now $\cot x = \frac{a}{h} \ , \cot y = \frac{a+1}{h} \ , \cot z = \frac{a+2}{h}$.
$$\frac{2}{\cot^2x - 2\cot^2y + \cot^2z} = \frac{2}{\frac{a^2}{h^2}-2\frac{a^2+2a+1}{h^2}+\frac{a^2+4a+4}{h^2}} = \frac{2h^2}{a^2-2a^2-4a-2+a^2+4a+4} = \frac{2h^2}{2}$$ 
Thus,
$$\sqrt{\frac{2}{\cot^2x - 2\cot^2y + \cot^2z}}  = h$$
</s>