In a $\triangle ABC$,the expression $(a-b)^2 \cos^2 \frac{C}{2} + (a+b)^2 \sin^2 \frac{C}{2}$ is equal to:

In a $\triangle ABC$,if $r_1=12, r_2=18$ and $r_3=36$,then $b=$

If the area of a triangle $ABC$ is $4\sqrt{5} \text{ sq. units}$, the length of the side $CA$ is $6 \text{ units}$, and $\tan \frac{B}{2} = \frac{\sqrt{5}}{4}$, then the length of its smallest side is: (in $\text{ units}$)

In a $\triangle ABC$ with $\angle A = 90^{\circ}$,$P$ is a point on $BC$ such that $PA:PB = 3:4$. If $AB = \sqrt{7}$ and $AC = \sqrt{5}$,then $BP:PC$ is

If one side of a triangle is double the other and the angles opposite to these sides differ by $60^{\circ}$,then the triangle is

In $\Delta ABC$,$\frac{\sin B}{\sin (A + B)} = $