In a $\Delta ABC,$ $2ac \sin \left( \frac{A - B + C}{2} \right)$ is equal to

In a $\triangle ABC$,$A = 30^{\circ} + C$ and $R = (\sqrt{3} + 1)r$,where $r$ is the inradius and $R$ is the circumradius. Then:

If $\Delta = a^2 - (b - c)^2$,where $\Delta$ is the area of triangle $ABC$,then $\tan A$ is equal to

If in $\Delta ABC$,$AB = 4$,$BC = 6$ and $AC = 5$,and $h_1, h_2, h_3$ are the lengths of the altitudes from vertices $A, B, C$ respectively,then the value of $(\frac{1}{h_1} + \frac{1}{h_2} - \frac{1}{h_3})$ is equal to-

In a triangle $ABC$,$\frac{a}{b}=2+\sqrt{3}$ and $\angle C=60^{\circ}$. Then the measure of $\angle A$ is (in $^{\circ}$)

In triangle $ABC$, with usual notations, if $(a + b + c)(a + b - c) = ab$, then the measure of angle $C$ is...