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The function $$f(x)=\cos(\sin(x))-\sin(\cos(x))$$ has a unique fix-point.

The solution of $\ f(x)=x\ $ is $$0.15328786038074973385826057\cdots$$

Is this number rational, algebraic irrational or transcendental ?

We cannot use the fact that $\cos(x)$ and $\sin(x)$ are transcendental for algebraic $x\ne 0$ because we have nested trigonometric functions and moreover a difference of such functions.

The continued fraction and the algdep-calculations with PARI/GP seem to indicate that the number is transcendental, but of course this is not a proof.

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    $\begingroup$ My guess is "transcendental." $\endgroup$ – Michael Sep 10 '16 at 23:08
  • $\begingroup$ The most probable result. But I have little hope that it can be proven. Even a proof that the number is irrational seems to be very difficult, if not impossible. $\endgroup$ – Peter Sep 10 '16 at 23:12
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    $\begingroup$ @Jam Anyway , thanks for the double sum. $\endgroup$ – Peter Sep 10 '16 at 23:20
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    $\begingroup$ @Peter Fool that I am, there was a mistake in the sum so I've taken it down anyway. C'est la vie. $\endgroup$ – Jam Sep 10 '16 at 23:29

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