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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.

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    Look it up in a table. If just this one value is at stake and it is not $0.000000013$ then anyone could do an estimate with pencil and paper if (s)he is not convinced.2012-04-15

2 Answers 2

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Recall that 1-\frac{1}{2}x^2 \;<\; \cos x \;<\; 1-\frac{1}{2}x^2 + \frac{1}{24}x^4 for $x>0$. From the upper bound, we see that \cos 1 < 13/24. Since $\cos$ is decreasing, it follows that $ \cos(\cos 1) \;>\; \cos\left(\frac{13}{24}\right) \;>\; 1-\frac{1}{2}\left(\frac{13}{24}\right)^2 \;>\; 0.85. $ Next, recall that \sin x < x for all $x>0$. Since $\sin x$ is increasing, this gives us \sin\sin\sin 1 \;<\; \sin \sin 1 \;<\; \sin 1. But we also know that \sin x \;<\; x - \frac{1}{6}x^3 + \frac{1}{120}x^5 for all $x>0$. It follows that \sin\sin\sin 1 \;<\; \sin 1 \;<\; \frac{101}{120} \;<\; 0.85.

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    I think the OP is trying to do this without a calculator.2016-06-30
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I guess you have written your question as you want it, since \cos\cos\cos 1= 0.65< \sin \sin \sin 1=0.67

If you really want only two $\cos$ iterations in the left then the inequality is true and you can prove it like this:

$\cos(\cos(1))=\sin(\pi/2-\cos 1)>\sin 1$

since $f(x)=\cos x+x$ is strictly increasing and $f(\pi/2)=\pi/2$. Therefore f(1)<\pi/2 and using the monotonicity of $\sin $ on $[0,\pi/2]$ you are done.

It remains to notice that $ \sin 1 >\sin \sin 1>\sin \sin \sin 1$