Question

A helicopter is flying at a height of 1000 feet above a small mountain peak as show in the figure below. The smaller mountain peak has an altitude of 6200 feet. A second, taller peak is viewed from both the smaller mountaintop and the helicopter. From the helicopter, the angle of depression is 43°, and from the smaller mountaintop, the angle of depression is 18°. Approximate the altitude of the taller peak and round your answer to the nearest foot.

Answer 1

check your wording:
... from the smaller mountaintop, the angle of depression is 18°

If you are looking from the smaller to the larger, it would be an angle of elevation, not depression.
I will assume that case.

Draw a diagram, draw a horizontal from the taller mountain
cutting the 1000 ft into x and 1000-x, the x close to the small mountain top
Let that horizontal distance be h
You now have two right-angled triangles so that
h/x = cot43° and h/(1000-x) = cot 18°
h = xcot43 and h = (1000-x)cot18

xcot43 = (1000-x)cot18
xcot43 + xcot18 = 1000
x = 1000/(cot43 + cot18) = ..... use your calculator, remember to get cotx, do 1/tanx

Look at your diagram, what does the x represent?