Is there any way to prove this? When I check intuitively by taking $A=0$ and $B=C=\frac{\pi}{2}$, the value of the expression becomes $2$ and as I changed the angles the value kept increasing in $0$ to $\frac{\pi}{2}$. I tried using Jensen's inequality but we get $\sin A + \sin B + \sin C <\frac{3\sqrt{3}}{2}$, which is of no use.