I proved the inequality shown in the title by using AM-GM and:
$$\cos A + \cos B + \cos C \le \frac{3}{2}$$
Equality holds for an equilateral triangle.

For an $n$-sided convex polygon with internal angles $\alpha\_1, \alpha\_2, \dots, \alpha\_n$, it appears to me that $\prod^n\_{k=1}1 - \cos\alpha\_k$ is maximum when $\alpha\_1 = \alpha\_2 = \dots = \alpha\_n$, i.e. when $\alpha\_i = \frac{(n-2)\pi}{n}$, thus
$$\prod^n\_{k=1}1 - \cos(\alpha\_k) \le \cos^n \pi\left(1-\frac{2}{n}\right)$$
How can this inequality be proved or disproved?