In any triangle $ABC$,$(b+c) \cos A + (c+a) \cos B + (a+b) \cos C$ equals

In any $\triangle ABC$,the simplified form of $\frac{\cos 2A}{a^2} - \frac{\cos 2B}{b^2}$ is

In a triangle $ABC$,with usual notations,if $a=5$,$b=7$,and $\sin A=\frac{3}{4}$,then the total number of triangles possible is:

In a triangle $ABC$, with the usual notations, $\angle B = \frac{\pi}{3}$ and $\angle C = \frac{\pi}{4}$. If $D$ divides $BC$ internally in the ratio $1:3$, then $\frac{\sin \angle BAD}{\sin \angle CAD} =$

In $\Delta ABC, a(b\cos C - c\cos B) = $

In a triangle $ABC$,let $AB = \sqrt{23}$,$BC = 3$,and $CA = 4$. Then the value of $\frac{\cot A + \cot C}{\cot B}$ is: