Can You Prove $\cos(\cos 1) > \sin(\sin(\sin 1))$?

[anemone]

Here is this week's POTW:

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Prove $\cos (\cos 1) > \sin (\sin (\sin 1))$.

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Remember to read the https://mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to https://mathhelpboards.com/forms.php?do=form&fid=2!

 

[anemone]

No one answered last week's POTW(Sadface), but you can find the suggested solution as follows:

Spoiler

For $x>0$, we have

$1-\dfrac{x^2}{2}< \cos x<1-\dfrac{x^2}{2}+\dfrac{x^4}{24}$

When $x=1$, we see that

$\cos 1<1-\dfrac{1^2}{2}+\dfrac{1^4}{24}=\dfrac{13}{24}$

Since $\cos x$ is decreasing. it follows that

$\cos (\cos 1)>\cos \left(\dfrac{13}{24}\right)>1-\dfrac{1}{2}\left(\dfrac{13}{24}\right)^2>0.85$

Next, recall that $\sin x<x$ for $x>0$.

Since $\sin x$ is increasing, we have

$\sin (\sin (\sin 1))<\sin (\sin 1)< \sin 1$

But we also know that

$\sin x<x-\dfrac{x^3}{6}+\dfrac{x^5}{120}$ for $x>0$.

It follows that

$\sin (\sin (\sin 1))< \sin 1<\dfrac{101}{120}<0.85<\cos (\cos 1)$ (Q.E.D.)