I would like to show that:
$ 1<\sin\frac{\alpha}{2}+\sin\frac{\beta}{2}+\sin\frac{\gamma}{2}$
where $\alpha, \beta, \gamma$ are the angles of a triangle.
I know that the inequality $ 1<\cos \alpha+\cos \beta+\cos \alpha $
is a direct consequence of the identity $ \cos \alpha+\cos \beta+\cos \alpha =1+\frac{r}{R}$
with circumradius $R$ and inradius $r$.
So is there a similar expression for $ \sin\frac{\alpha}{2}+\sin\frac{\beta}{2}+\sin\frac{\gamma}{2}?$