This is the problem given: The side of a hill faces due south and is inclined to the horizontal at an angle alpha. A straight railway upon it is inclined at an angle beta to the horizontal. If the bearing of the railway is gamma east of north, show that cos(gamma) = cot(alpha)tan(beta)

Question

I drew a diagram like this, according to this question (imageshack.us/photo/my-images/30/trigonometryquestion.png/).
We have cos(gamma) = CH/AC
cot(alpha) = BH/CH
tan(beta) = CH/AH,
so cot(alpha)xtan(beta) = BH/AH
I'm stuck at proving CH/AC = BH/AH

Is there anyway around this? I'll appreciate any help from you all. Thank you very much.

Note: AC is not perpendicular to BC.

Answer 1

From the diagram, I am not sure if you have the same interpretation of the question as I have.

There is a slope facing south, at α with the horizontal.
The railway is at an angle of γ east of north (on the horizontal projection).
The angle β is the angle the railway makes with the horizontal.

If you look at the situation in plan view, north towards the top of the paper, we see a line at γ towards east. Let the railway line length be L, and denote A by the south end of the line, and B the north end of the line.

Let the elevation of A (south) be zero.
Then B is h above A, where
h=Lsinβ.

Now we will calculate h in a different way by projecting the point B to a north-south line on the slope, call it B'.

B' should be also h above point A, since the side of the hill faces north-south.

Now calculate h by first projecting L onto the horizontal plane, then project the resulting line to the N-S line, and finally multiply by tanα to get the height h.
h=Lcosβcosγtanα

Thus
h=Lsinβ=Lcosβcosγtanα
transpose cosγ to the left, and cancel common factor L, we get:
cosγ=(sinβ/cosβ)cotα
=tanβcotα. QED