<s>[INST] Minimum value of cosA+cosB+cosC in a triangle ABC I have used jensen's inequality but couldn't move on.
 [/INST] $$
\begin{align}
\sqrt[3]{\sin(\frac{A}{2}) \sin(\frac{B}{2}) \sin(\frac{C}{2})} 
&
\le \frac{\sin(\frac{A}{2}) + \sin(\frac{B}{2}) + \sin(\frac{C}{2}) }{3} \text{  (AM-GM inequality)}\\
&
\le \sin \left( \frac{ \frac{A}{2} + \frac{B}{2} + \frac{C}{2}}{3} \right) \text{ (Jensen's inequality)}\\
&
= \sin(\frac{\pi}{6})\\
&
= \frac{1}{2}
\end{align}
$$
$$
0 \lt \sin(\frac{A}{2}) \sin(\frac{B}{2}) \sin(\frac{C}{2}) \le  \frac{1}{8} 
$$
since
$$
\cos A+\cos B+\cos C=1+4\sin(\frac{A}{2}) \sin(\frac{B}{2}) \sin(\frac{C}{2})
$$
so we have:
$$
1 \lt \cos A+\cos B+\cos C \le \frac{3}{2}
$$
</s>