<s>[INST] Let $\alpha$, $\beta$, and $\gamma$ be acute angles such that $\alpha$ + $\beta$ = $\gamma$. Show that Let $\alpha$, $\beta$, and $\gamma$ be acute angles such that $\alpha$ + $\beta$ = $\gamma$. Show that
cos$\alpha$ + cos$\beta$ + cos$\gamma$ - 1 $\geq$ $2\sqrt{\cos\alpha\times\cos\beta\times\cos\gamma}$.
Here's what I've tried. I know that $0 < \alpha, \beta, \gamma < \pi/2$ since they are acute angles. So I substituted $\alpha+\beta$ for $\gamma$ 
$\cos\alpha$ + $\cos\beta$ + $\cos(\alpha+\beta)$- 1 $\geq$ $2\sqrt{\cos\alpha\cos\beta\cos(\alpha+\beta)}$.
$\cos\alpha + \cos\beta + \cos(\alpha)\cos(\beta)-\sin(\alpha)\sin(\beta)- 1 \geq 2\sqrt{\frac{1}{2}[\cos(\alpha-\beta)+\cos(\alpha+\beta)]\cos(\alpha+\beta)}.$
$\cos\alpha + \cos\beta + \cos(\alpha)\cos(\beta)-\sin(\alpha)\sin(\beta)- 1 \geq 2\sqrt{\frac{1}{2}[\cos(\alpha-\beta)\cos(\alpha+\beta)+\cos^{2}(\alpha+\beta)]}$
$\cos\alpha + \cos\beta + \cos(\alpha)\cos(\beta)-\sin(\alpha)\sin(\beta)- 1 \geq 2\sqrt{\frac{1}{4}[\cos(-2\beta)+\cos(2\alpha)]+\cos^{2}(\alpha+\beta)}$
I got stuck from there and I'm not sure where to go from here. Any help would be appreciated.
 [/INST] Using that
$$
\cos\alpha\cos\beta-\cos\gamma = \sin\alpha\sin\beta
 = 4 \sin\frac\alpha2 \cos\frac\alpha2 \sin\frac\beta2 \cos\frac\beta2 \ge 0
$$
and
$$
(1-\cos\alpha)(1-\cos\beta) = 4 \sin^2\frac\alpha2 \sin^2\frac\beta2,
$$
a possible proof:
$$
\cos\gamma - 2\sqrt{\cos\alpha\cos\beta}\cdot\sqrt{\cos\gamma}+
\cos\alpha\cos\beta
\stackrel{?}{\ge} 
(1-\cos\alpha)(1-\cos\beta) 
$$
$$
\Big(\underbrace{\sqrt{\cos\alpha\cos\beta}-\sqrt{\cos\gamma}}_{\ge0}\Big)^2
\stackrel{?}{\ge} 
4\sin^2\frac\alpha2 \sin^2\frac\beta2
$$
$$
\sqrt{\cos\alpha\cos\beta}-\sqrt{\cos\gamma}
\stackrel{?}{\ge} 
2 \sin\frac\alpha2 \sin\frac\beta2
$$
$$
\cos\alpha\cos\beta 
\stackrel{?}{\ge} 
\Big(\sqrt{\cos\gamma} + 2 \sin\frac\alpha2 \sin\frac\beta2\Big)^2
$$
$$
\cos\alpha\cos\beta-\cos\gamma
\stackrel{?}{\ge} 
4 \sin\frac\alpha2 \sin\frac\beta2 \sqrt{\cos\gamma} 
+ 4 \sin^2\frac\alpha2 \sin^2\frac\beta2
$$
$$
4 \sin\frac\alpha2 \cos\frac\alpha2 \sin\frac\beta2 \cos\frac\beta2 \stackrel{?}{\ge} 
4 \sin\frac\alpha2 \sin\frac\beta2 \sqrt{\cos\gamma} 
+ 4 \sin^2\frac\alpha2 \sin^2\frac\beta2
$$
$$
 \cos\frac\alpha2 \cos\frac\beta2
-  \sin\frac\alpha2 \sin\frac\beta2
\stackrel{?}{\ge} 
\sqrt{\cos\gamma} 
$$
$$
\cos\frac\gamma2 \stackrel{?}{\ge} 
\sqrt{\cos\gamma} 
$$
$$
\sqrt{\frac{1+\cos\gamma}2} \stackrel{?}{\ge} 
\sqrt{\cos\gamma} 
$$
$$
1 \ge \cos\gamma.
$$
</s>