In $\triangle ABC$,find the value of $r r_1 \cot \frac{A}{2} + r r_2 \cot \frac{B}{2} + r r_3 \cot \frac{C}{2}$.

In a triangle $ABC$,if $a^2-b^2-c^2=bc(\lambda^2-2\lambda-1)$,then

With usual notation in a $\Delta ABC$,$\left( \frac{1}{r_1} + \frac{1}{r_2} \right) \left( \frac{1}{r_2} + \frac{1}{r_3} \right) \left( \frac{1}{r_3} + \frac{1}{r_1} \right) = \frac{K R^3}{a^2 b^2 c^2}$ where $K$ has the value equal to

The common solution set of the equations $2 \sin^2 x + \sin^2 2x = 2$ and $\sin 2x + \cos 2x = \tan x$ is

In $\triangle PQR$,if $\angle R = \frac{\pi}{4}$ and $\tan(\frac{P}{3})$,$\tan(\frac{Q}{3})$ are the roots of the equation $ax^2 + bx + c = 0$,then:

In $\triangle ABC$,if $\angle C=90^{\circ}$,then $\left(\frac{r_1-r_3}{r_1}\right)\left(\frac{r_2-r_3}{r_2}\right)=$