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Homework Statement
A kite 100 ft above the ground moves horizontally at a speed of 8 ft/s. At what rate is the angle between the string and the horizontal decreasing when 200ft of the string has been let out?
Homework Equations

The Attempt at a Solution
I've tried two different methods to come up with the answer but i'm not sure which one is correct.
In one case, I've got:
cosθ = x/200
And in the other one:
tanθ = 100/x
However, when I try to derive both sides of both equations in order to get the result, I found out that the values of dθ/dt are actually different for each equation. For example, in the first case:
sinθ (dθ/dt) = 1/200 (dx/dt)
Clearing this out makes the final result as (2/25)
On the other hand:
sec^{2} (dθ/dt) = (dx/dt) / 3
Clearing this out makes the final result as (1/50)
Is there a reason for this incongruity or have I made a mistake in my calculations? Thank you very much guys.