<s>[INST] Minimum value of cosA+cosB+cosC in a triangle ABC I have used jensen's inequality but couldn't move on.
 [/INST] $\cos A+\cos B+\cos C$
$=2\cos\dfrac{A+B}2\cos\dfrac{A-B}2+1-2\sin^2\dfrac C2$
$=2\sin\dfrac C2\cos\dfrac{A-B}2+1-2\sin\dfrac C2\cos\dfrac{A+B}2$ as $A+B=\pi-C,\dfrac{A+B}2=\dfrac\pi2-\dfrac C2$
$=1+2\sin\dfrac C2\left[\cos\dfrac{A-B}2-\cos\dfrac{A+B}2\right]$
$=1+4\sin\dfrac A2\sin\dfrac B2\sin\dfrac C2$
Now $\sin\dfrac A2\sin\dfrac B2\sin\dfrac C2$ can be made arbitrarily small, but$>0$ by setting $A,B\to0,C\to\pi$
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