In a triangle $ABC$,if $a \neq b$,then the value of $\frac{a \cos A - b \cos B}{a \cos B - b \cos A} + \cos C$ is:

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

If $a=3, b=5, c=7$ are the sides of a triangle $ABC$,then $\cot A+\cot B+\cot C=$

If $p_1, p_2, p_3$ are altitudes of a triangle $ABC$ from the vertices $A, B, C$ respectively,and $\Delta$ is the area of the triangle,then $p_1^{-1} + p_2^{-1} - p_3^{-1}$ is equal to

If,in a $\triangle ABC$,$\tan \frac{A}{2} = \frac{5}{6}$ and $\tan \frac{C}{2} = \frac{2}{5}$,then $a, b, c$ are such that :

In a $\triangle ABC$,if $a+c=5b$,then $\cot \frac{A}{2} \cot \frac{C}{2}$ is equal to