In $\triangle ABC$,find the value of $\frac{1+\cos C}{r_1+r_2}+\frac{1+\cos A}{r_2+r_3}+\frac{1+\cos B}{r_1+r_3}$.

In a triangle $ABC$,if $\frac{a}{\tan A}=\frac{b}{\tan B}=\frac{c}{\tan C}$,then $\cos^2 A + \cos^2 B + \cos^2 C =$

If the radius of the circumcircle of an isosceles triangle $PQR$ is equal to $PQ$ (where $PQ = PR$),then the angle $P$ is

In a $\triangle ABC$,$\angle C = 60^{\circ}$ and $\angle A = 75^{\circ}$. If $D$ is a point on $AC$ such that the area of $\triangle BAD$ is $\sqrt{3}$ times the area of $\triangle BCD$,then the measure of $\angle ABD$ is (in $^{\circ}$)

In $\triangle ABC$,if $(\sin A+\sin B)(\sin A-\sin B)=\sin C(\sin B+\sin C)$,then $\angle A=$ (in $^{\circ}$)

In $\triangle ABC$,$\angle B=60^{\circ}$ and $\angle A=75^{\circ}$. If a point $D$ divides $BC$ in the ratio $2:3$,then $\sin \angle BAD : \sin \angle CAD=$