In $\triangle ABC$,if $a=5$ and $\tan \frac{A-B}{2}=\frac{1}{4} \tan \frac{A+B}{2}$,then $\sqrt{a^2-b^2}=$

In a triangle $ABC$,if $B = 3C$,then the values of $\sqrt{\frac{b + c}{4c}}$ and $\frac{b - c}{2c}$ are

If in a right-angled triangle the hypotenuse is four times as long as the perpendicular drawn to it from the opposite vertex,then one of its acute angles is......$^o$

In a triangle $ABC$,if $A=30^{\circ}$ and $\frac{b}{(\sqrt{3}+1)^2+2(\sqrt{2}-1)}=\frac{c}{(\sqrt{3}+1)^2-2(\sqrt{2}-1)}$,then $B=$ (in $^{\circ}$)

In a $\Delta ABC$,if two sides $b$ and $c$ and an angle $B$ are given.
Statement $-1$: Let $b = 5 \ cm, c = 3 \ cm$ and $\angle B = 60^\circ$. The total number of possible triangles is $2$.
Statement $-2$: If $c \sin B < b < c$ and $B$ is an acute angle,then there are two possible values of $\angle C$.

In a triangle $ABC$,with usual notations,if $\tan \left(\frac{A}{2}\right) = \frac{5}{6}$ and $\tan \left(\frac{C}{2}\right) = \frac{2}{5}$,then: