Trigonometry! #103

We know that $AM\ge GM$ , and the minimum value of the arithmetic mean is obtained when $AM=GM$ , which means that the terms are equal.

$\therefore \cos \left(\alpha+\frac{\pi}{2}\right)= \cos \left(\beta+\frac{\pi}{2}\right)= \cos \left(\gamma+\frac{\pi}{2}\right)$

$\Rightarrow\alpha=\beta=\gamma$

As $\alpha+\beta+\gamma=2\pi$ , we get $\alpha=\beta=\gamma=\frac{2\pi}{3}$

Hence $\frac{ \cos \left(\alpha+\frac{\pi}{2}\right)+ \cos \left(\beta+\frac{\pi}{2}\right)+ \cos \left(\gamma+\frac{\pi}{2}\right)} {3}$

$=\frac{ \cos \left(\frac{2\pi}{3} +\frac{\pi}{2}\right)+ \cos \left(\frac{2\pi}{3}+\frac{\pi}{2}\right)+ \cos \left(\frac{2\pi}{3}+\frac{\pi}{2}\right)} {3}$

$=-\sin\left(\frac{2\pi}{3}\right)$

$=\boxed{-\frac{\sqrt{3}}{2}}$