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 $$ \sin\frac{\alpha}{2}+\sin\frac{\beta}{2}+\sin\frac{\gamma}{2}?$$