In a $\triangle ABC$,if $4a = b + c$,then $\tan \frac{B}{2} \tan \frac{C}{2} =$

In $\triangle ABC$,$\angle B=60^{\circ}$ and $\angle A=75^{\circ}$. If a point $D$ divides $BC$ in the ratio $2:3$,then $\sin \angle BAD : \sin \angle CAD=$

In a triangle $ABC$,$AD$ is the altitude from $A$. Given $b > c$,$\angle C = 23^\circ$ and $AD = \frac{abc}{b^2 - c^2}$,then $\angle B = $ .....$^\circ$

In a triangle $ABC$ with usual notations $a=2$ and $b=3$,the value of $\frac{\cos 2A}{a^2} - \frac{\cos 2B}{b^2}$ is:

In $\Delta ABC$,$8 \Delta = (b + c)(bc + 1)$,then the circumradius of $\Delta ABC$ is (where $\Delta$ denotes the area of the triangle and $b, c$ are the lengths of sides $AC$ and $AB$ respectively):

In a triangle $ABC$,$(b+c) \sin \frac{A}{2} =$