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.

  • $\begingroup$ If there is a more appropriate tag please feel free to suggest :) $\endgroup$ Apr 15 '12 at 18:34
  • $\begingroup$ Have you tried to calculate the values with a computer calculator? If yes, then verifying the computation shouldn't be such a difficult task. $\endgroup$ Apr 15 '12 at 19:16
  • $\begingroup$ 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. $\endgroup$ Apr 15 '12 at 19:20

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 $$ \sin\sin\sin 1 \;<\; \sin 1 \;<\; \frac{101}{120} \;<\; 0.85. $$

  • 4
    $\begingroup$ Of course, this kind of estimation proof isn't really necessary. If you want to know whether $\cos\cos 1$ or $\sin\sin\sin 1$ is bigger, you can just plug them into a calculator and get an answer. $\endgroup$
    – Jim Belk
    Apr 15 '12 at 19:26
  • 1
    $\begingroup$ I think the OP is trying to do this without a calculator. $\endgroup$ Jun 30 '16 at 22:17

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 $$ \sin 1 >\sin \sin 1>\sin \sin \sin 1$$


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