# How prove this inequality sinsinsinsinx≤45coscoscoscosx\sin{\sin{\sin{\sin{x}}}}\le\frac{4}{5}\cos{\cos{\cos{\cos{x}}}}

Nice Question:

let $x\in [0,2\pi]$, show that:

I know this follow famous problem(1995 Russia Mathematical olympiad)

This problem solution can see :http://iask.games.sina.com.cn/b/19776980.html
and everywhere have solution in china BBS

I post this problem solution

case1: if $x\in[\pi,2\pi]$,then

so

case2: if $x\in[0,\dfrac{\pi}{2}]$,then we have

so

then

so

so

then

case3: if $x\in (\dfrac{\pi}{2},\pi)$,then let

$y=x-\dfrac{\pi}{2}$,so

and since
$f(t)=\sin{\sin{t}}$ is increasing,then

so

so

But I found this $\dfrac{4}{5}$ maybe is strong,

so if $x\in[\pi,2\pi]$,then we have

But for the case $x\in [0,\pi]$, I can't prove this

Thank you very much！

We still start from the original Russian Olympiad Problem:
$\cos \cos \cos \cos x> \sin \sin \sin \sin x$. It could have another numerical proof simply by doing in a calculator:

We have $-1\leq \cos x \leq 1, \text{that is }\cos 1\leq \cos \cos x\leq 1, \text{that is }\cos 1 \leq \cos \cos \cos x \leq \cos \cos 1$.

Finally, we have,

Similarly, we have

If the equation has the solution, that is

Thus, we have

Therefore, $\cos \cos \geq 0.7013 \to \cos \cos \cos x \leq 0.7639 \to \cos \cos \cos \cos \cos x \geq 0.7221...$

Thus, it is not possible to have $\sin \sin \sin \sin \sin \leq 0.6784.$ The inequality holds.

So, the $\frac{4}{5}$ is still not a strong constant, and inequality proof as similar.