In $\Delta ABC$,if $a = 3, b = 4, c = 5$,then $\sin 2B = $

In triangle $ABC,$ $\frac{1 + \cos(A - B)\cos C}{1 + \cos(A - C)\cos B} = $

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:

If $b+c=3a$,then $\cot \frac{B}{2} \cot \frac{C}{2}$ is equal to :

With usual notations,if the angles of a triangle are in the ratio $1: 2: 3$,then their corresponding sides are in the ratio.

In $\triangle ABC$,$(r_1+r_2) \operatorname{cosec}^2 \frac{C}{2} =$