<s>[INST] Prove $8(1-\cos^2a)(1-\cos^2b)(1-\cos^2c) \geq 27 \cos a\cos b\cos c$ 
Let $a$, $b$ and $c$ be angles of an acute triangle. Prove that:
  $$8(1-\cos^2a)(1-\cos^2b)(1-\cos^2c) \geq 27 \cos a\cos b\cos c$$

I tried am-gm, rm-gm rm-am and a couple of other inequalities but I didn't get anywhere.
 [/INST] Here's a proof that accounts for all acute-angled triangles whose angles lie in $ \left(\tan^{-1}{\sqrt{2}}, \dfrac{\pi}{2}\right)$.
Note that $f(x) = \ln \sin(x) \tan(x)$ is convex over $ \left( \tan^{-1}{\sqrt{2}}, \dfrac{\pi}{2}\right)$ since $f''(x) = \dfrac{\tan^{2}(x) -2}{\sin^{2}(x)} > 0 \quad \forall x\in \left(\tan^{-1}{\sqrt{2}}, \dfrac{\pi}{2}\right)$
Hence we can apply Jensen's inequality to $f(x)$ and $a, b,c \in \left(\tan^{-1}{\sqrt{2}}, \dfrac{\pi}{2}\right)$ to get:
$f\left(\dfrac{a+b+c}{3}\right)\leq \dfrac{f(a) + f(b)+ f(c)}{3}$
$\Rightarrow \ln \dfrac{3}{2} \leq \dfrac{\ln \dfrac{\sin^{2}(a)\,\sin^{2}(b)\,\sin^{2}(c)}{\cos(a)\,\cos(b)\,\cos(c)}}{3}$
$\Rightarrow \dfrac{27}{8} \leq \dfrac{\sin^{2}(a)\,\sin^{2}(b)\,\sin^{2}(c)}{\cos(a)\,\cos(b)\,\cos(c)}$, as required.
Can someone figure out an extension of this argument that accounts for those acute-angled triangles with one angle in $ \left(0, \tan^{-1}{\sqrt{2}}\right)$?
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