<s>[INST] Inequality on tangent and secant function 
Let $\{\alpha, \beta, \gamma, \delta\} \subset \left (0,\frac {\pi}{2}\right)$ and $ \alpha + \beta+\gamma+\delta = {\pi}$. Prove that $\sqrt {2} \left(\tan \alpha +\tan \beta +\tan\gamma+\tan \delta\right)\ge \sec \alpha +\sec \beta +\sec \gamma +\sec \delta$

 [/INST] Let $f(x)=\sqrt2\tan{x}-\frac{1}{\cos{x}}$.
Thus, $$f''(x)=\frac{(1+\sqrt2-\sin{x})(\sin{x}-\sqrt2+1)}{\cos^3x}>0$$
for $x\in\left[\frac{\pi}{4},\frac{\pi}{2}\right)$.
Thus, by Vasc's RCF Theorem it's enough to prove our inequality for
$\beta=\gamma=\alpha$ and $\delta=\pi-3\alpha$, where $\alpha\in\left[\frac{\pi}{4},\frac{\pi}{3}\right)$ and the rest is smooth.
Indeed, we need to prove that
$$\frac{3(\sqrt2\sin{\alpha}-1)}{\cos{\alpha}}+\frac{\sqrt{2}\sin\left(180^{\circ}-3\alpha\right)-1}{\cos\left(180^{\circ}-3\alpha\right)}\geq0.$$
Let $\sin\alpha=t$.
Thus, $\frac{1}{\sqrt2}\leq t<\frac{\sqrt3}{2}$ and we need to prove that
$$\frac{3(\sqrt2t-1)}{\cos\alpha}-\frac{\sqrt2\sin3\alpha-1}{\cos3\alpha}\geq0$$ or
$$3(\sqrt2t-1)-\frac{\sqrt2(3t-4t^3)-1}{4\cos^2\alpha-3}\geq0$$ or
$$3(\sqrt2t-1)-\frac{\sqrt2(3t-4t^3)-1}{4(1-t^2)-3}\geq0$$ or
$$3(\sqrt2t-1)+\frac{\sqrt2(3t-4t^3)-1}{4t^2-1}\geq0$$ or
$$4\sqrt2t^3-6t^2+1\geq0,$$
which is true by AM-GM:
$$4\sqrt2t^3+1=2\sqrt2t^3+2\sqrt2t^3+1\geq3\sqrt[3]{\left(2\sqrt2t^3\right)^2\cdot1}=6t^2.$$
Done!
We can use also the Tangent Line method. 
$$\sum_{cyc}\left(\sqrt2\tan\alpha-\sec\alpha\right)=\sum_{cyc}\left(\sqrt2\tan\alpha-\sec\alpha-\sqrt2\left(\alpha-\frac{\pi}{4}\right)\right)\geq0$$
because $f(x)\geq0$ for all $0<x<\frac{\pi}{2}$, where
$$f(x)=\sqrt2\tan{x}-\sec{x}-\sqrt2\left(x-\frac{\pi}{4}\right).$$
Indeed, $$f'(x)=\frac{\sqrt2}{\cos^2x}-\frac{\sin{x}}{\cos^2x}-\sqrt2=\frac{\sin{x}(\sqrt2\sin{x}-1)}{\cos^2x},$$
which gives $x_{min}=\frac{\pi}{4}.$
Id est, $$f(x)\geq f\left(\frac{\pi}{4}\right)=0$$
and we are done!
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