I'm proud to present one inequality of my work :
Let $a,b,c>0$ such that $a+b+c=1$ then we have : $$3\cos(\frac{1}{3}\tan(\frac{1}{3}\sin(\frac{1}{3})))\leq\sum_{cyc}\cos(a\tan(b\sin(c)))\leq 3$$
The goal was to create an inequality where the value are very condensed because we have :
$$3-3\cos(\frac{1}{3}\tan(\frac{1}{3}\sin(\frac{1}{3})))\leq 0.002$$
The RHS is obvious since $\cos(x)\leq 1 $ $ \forall x\in(-\infty,\infty)$
I have try power series but I'm stuck because the inequality is cyclic .
I think we can use Am-Gm and try to take the logarithm on each side but with this method I get nothing good .
Finally I try to use derivatives but it beginns very complicated since we composed elementary function .
If you have just a hint or a pedagogic method it would be cool.
Thanks a lot for sharing your time and knowledge .