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If A, B, and C are the interior angles of a triangle, then

cosA + CosB + cosC <= 3/2.

The inequality on the right can be proven by, first reducing it to two angles and then differentiation. There is only one local extremum attained for A = B = C = π/3.

You may be curious to have a look at the old CTK Exchange archive.Please do not post there.

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