If $\Delta$ denotes the area of $\triangle A B C$, then $(b \sin C+c \sin B)(b \cos C+c \cos B)=$

Let $A$ be the area of in-circle and $A_1, A_2, A_3$ be the area of ex-circles of a triangle. If $A_1=4, A_2=9, A_3=16$, then $A=$

In a $\triangle A B C$, if $(b+c)^2 \sin ^2 \frac{A}{2}+(b-c)^2 \cos ^2 \frac{A}{2}=K(1-\cos 2 A)$, then $K=$

In a $\triangle A B C$, if $b=7, c=4 \sqrt{3}$ and $A=\frac{\pi}{6}$ then a $\sin B \sin C=$