This is not a puzzle but an observation of something that to me was surprising.
Both present an equation that does not have an analytical solution for a crucial angle, let's call it x.
The equation that presents itself is this:
tan(x) = x + k
where k is some constant, and x is expressed in radians.
You can't get a general solution x = x(k), but you can solve the equation iteratively for x for any particular value of k.
When that is done, the the desired answer turns out to be
cos(x) = answer.
I wonder whether there is a prototypical problem for which this is the best analysis?
Did this post turn out to be a puzzle after all?