In a triangle $ABC$,$a[b \cos C - c \cos B] = $

With usual notations,in triangle $ABC$,$a=\sqrt{3}+1$,$b=\sqrt{3}-1$ and $m \angle C=60^{\circ}$,then $A-B=$ (in $^{\circ}$)

If in a $\triangle ABC$,$r_3 = r_1 + r_2 + r$,then $\angle A + \angle B$ is equal to (in $^{\circ}$)

In a $\Delta ABC$,if $b + c = 3a$,then the value of $\cot\, \frac{B}{2} \cdot \cot\, \frac{C}{2}$ is equal to:

If in a triangle $ABC$,$a=2$,$b=3$ and $c=4$,then $\tan \left(\frac{A}{2}\right) = $

In a triangle,the lengths of the sides are integers. Suppose that the length of one side is $1$,and the longest altitude is twice the shortest altitude. Let $R$ and $r$ be the circumradius and inradius of the triangle,respectively. If $R:r = m:n$,where $m$ and $n$ are coprime positive integers,then $m + n$ is