How can I show that
$\cos \cos 1 - \sin \sin \sin 1$
is positive?
This is motivated by this question. If
$\begin{align} f(x) &= \cos \cos \cos \cos(\pi/2 + ix) - \sin \sin \sin \sin(\pi/2+ix)\\ &= \cos \cos \cos \sinh x - \sin \sin \sin \cosh x, \end{align}$
then it looks like $f(x)$ has a zero in the interval $(0,1)$. This would imply that
$ \cos \cos \cos \cos(z) - \sin \sin \sin \sin(z) $
has infinitely-many zeros in the strip $0 < \Im(z) < 1$.
One way to show that such a zero exists would be to show that $f(1) < 0 < f(0)$, the right-side of which is the current question. I don't know how to show the left side either, but now I'm interested in this question for its own sake.