<s>[INST] How prove this inequality $\frac{e^x}{x+1}>\frac{\cos{x}}{\sin{x}+\sqrt{2}}$ Today,when I use wolf found this following inequality

let $x>-1$, show that
  $$\dfrac{e^x}{x+1}>\dfrac{\cos{x}}{\sin{x}+\sqrt{2}}$$

I found this 
I want $$\Longleftrightarrow (\sin{x}+\sqrt{2})e^x-(x+1)\cos{x}>0$$
let
$$f(x)=(\sin{x}+\sqrt{2})e^x-(x+1)\cos{x}\Longrightarrow f'(x)=\sqrt{2}e^x+(x+e^x+1)\sin{x}+(e^x-1)\cos{x}$$
then I calculus $f''(x)$ and found ugly,so maybe this inequality have other methods?
 [/INST] We know $\sqrt{2}+\sin x > 0$ since $|\sin x| \le 1$ and $\sqrt{2} > 1$.
$\displaystyle \frac{e^{x}}{x+1} \ge 1 \ge \frac{\cos x}{\sqrt{2}+\sin x}$, the later inequality is true since, 
$1 \ge \sin (\frac{\pi}{4} - x) = \dfrac{\cos x - \sin x}{\sqrt{2}}$
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