This is not a puzzle but an observation of something that to me was surprising.

Two unrelated puzzles - the oldie with a twist and the weight-loss problem, succumb to identical analyses.

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?