In a $\triangle A B C, 2 A+C=300^{\circ}$. If the circumradius of the $\triangle A B C$ is eight times its inradius, then $\sin \frac{C}{2}=$

In $\triangle A B C$, if $a=5, b=4$ and $\cos (A-B)=\frac{31}{32}$, then $c=$

In $\triangle A B C$, if $A, B, C$ are in arithmetic progression, then

$$ \sqrt{a^2-a c+c^2} \cdot \cos \left(\frac{A-C}{2}\right)= $$

If in $\triangle A B C, B=45^{\circ}, a=2(\sqrt{3}+1)$ and area of $\triangle A B C$ is $6+2 \sqrt{3}$ sq. units, then the side $b=$