> And a complication:
> In [17]: f(1,2)
> Out[17]:
>
>  log(cos(2) + 1)
>        log(-1 + cos(2))
> - ─────────────── + 2.99489845376757 - 0.5⋅ⅈ⋅π + ──────────────── > 1
>         2
>               2

The problem here seems to be in the rounding:

In [32]: a = log(cos(y) - 1)/2 - log(cos(y) + 1)/2 + 2.99489845376757 - 1j*pi/2

In [33]: a.subs(y, 2).evalf()
Out[33]: 3.43792117788449 - .0e-21⋅ⅈ

In [34]: a.subs(y,1.1).evalf()
Out[34]: 2.50567973267913

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