In a $\triangle ABC$,$\operatorname{cosec} A(\sin B \cos C + \cos B \sin C)$ is equal to

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

In any triangle $ABC$,$\frac{\tan \frac{A}{2} - \tan \frac{B}{2}}{\tan \frac{A}{2} + \tan \frac{B}{2}} = $

In a $\Delta ABC$,let $a, b$ and $c$ denote the lengths of sides opposite to vertices $A, B$ and $C$ respectively. If $b = 2, c = \sqrt{3}$ and $\angle BAC = \frac{\pi}{6}$,then the value of the circumradius of triangle $ABC$ is:

In $\triangle ABC$,with usual notations,the value of $\frac{b \sin B - c \sin C}{\sin (B - C)}$ is:

If in a triangle $ABC$,$\frac{\sin A}{4} = \frac{\sin B}{5} = \frac{\sin C}{6}$,then the value of $\cos A + \cos B + \cos C$ is equal to