<s>[INST] How prove this inequality $\sin{\sin{\sin{\sin{x}}}}\le\frac{4}{5}\cos{\cos{\cos{\cos{x}}}}$ Nice Question:
let $x\in [0,2\pi]$, show that:

$$\sin{\sin{\sin{\sin{x}}}}\le\dfrac{4}{5}\cos{\cos{\cos{\cos{x}}}}?$$

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

$$\sin{\sin{\sin{\sin{x}}}}<\cos{\cos{\cos{\cos{x}}}}$$

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
  $$\cos{\cos{\cos{\cos{x}}}}>0,\sin{\sin{\sin{\sin{x}}}}\le 0$$
  so
  $$\cos{\cos{\cos{\cos{x}}}}>\sin{\sin{\sin{\sin{x}}}}$$

case2: if $x\in[0,\dfrac{\pi}{2}]$,then we have
$$\cos{x}+\sin{x}\le\sqrt{2}<\dfrac{\pi}{2}\Longrightarrow 0\le \cos{x}<\dfrac{\pi}{2}-\sin{x}$$
so
$$\cos{\cos{x}}>\cos{\left(\dfrac{\pi}{2}-\sin{x}\right)}=\sin{\sin{x}}$$
$$\sin{\cos{x}}<\sin{\left(\dfrac{\pi}{2}-\sin{x}\right)}=\cos{\sin{x}}$$
then
$$\cos{\cos{\cos{x}}}<\cos{\sin{\sin{x}}}$$
so
$$\cos{\cos{\cos{x}}}+\sin{\sin{\sin{x}}}<\cos{\sin{\sin{x}}}+\sin{\sin{\sin{x}}}<\dfrac{\pi}{2}$$
so
$$\cos{\cos{\cos{x}}}<\dfrac{\pi}{2}-\sin{\sin{\sin{x}}}$$
then
$$\cos{\cos{\cos{\cos{x}}}}>\cos{\left(\dfrac{\pi}{2}-\sin{\sin{\sin{x}}}\right)}=\sin{\sin{\sin{\sin{x}}}}$$
case3: if $x\in (\dfrac{\pi}{2},\pi)$,then let

$y=x-\dfrac{\pi}{2}$,so
  $$\cos{\cos{\cos{\sin{y}}}}>\sin{\sin{\cos{\sin{y}}}}$$
  and since
  $f(t)=\sin{\sin{t}}$ is increasing,then
  $$f(\cos{\sin{y}})>f(\sin{\cos{y}})\Longrightarrow \sin{\sin{\cos{\sin{y}}}}>\sin{\sin{\sin{\cos{y}}}}$$
  so
  $$\cos{\cos{\cos{\sin{y}}}}>\sin{\sin{\sin{\cos{y}}}}$$
  so
  $$\cos{\cos{\cos{\cos{x}}}}>\sin{\sin{\sin{\sin{x}}}}$$

But I found this $\dfrac{4}{5}$ maybe is strong,
so if $x\in[\pi,2\pi]$,then we have
$$\dfrac{4}{5}\cos{\cos{\cos{\cos{x}}}}\ge 0>\sin{\sin{\sin{\sin{x}}}}$$
But for the  case  $x\in [0,\pi]$, I can't prove this
$$4\cos{\cos{\cos{\cos{x}}}}\ge 5\sin{\sin{\sin{\sin{x}}}}$$
Thank you very much！
 [/INST] 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,
$$
~0.6542 \simeq \cos \cos \cos 1\leq \cos \cos \cos \cos x \leq \cos \cos 1.
$$
Similarly, we have 
$$
-\sin \sin \sin \sin 1 \leq \sin \sin \sin \sin x \leq \sin \sin \sin 1 \simeq0.6784...
$$
If the equation has the solution, that is
$$
\sin x\geq \sin^{-1} \sin^{-1}\sin ^{-1} \cos \cos \cos 1 \simeq 0.6086...
$$
Thus, we have 
$$
|\cos x|\leq 0.7835...
$$
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.
</s>