If $A, B, C$ are the angles of a triangle,then $\sum \frac{\cot A + \cot B}{\tan A + \tan B} = $

$A$ triangle $ABC$ has area of $P$ square units and perimeter $2S$ units. If $h_1, h_2$ and $h_3$ are respectively the lengths of the altitudes of the triangle drawn from the vertices $A, B$ and $C$,then $P^2 \left[ \frac{(h_1 h_2 + h_2 h_3 + h_3 h_1)^2}{h_1^2 h_2^2 h_3^2} - 2 \right] =$

If $x, y, z$ are the lengths of the perpendiculars drawn from the circumcenter to the sides $a, b, c$ respectively of a triangle,then the value of $\frac{bx}{c} + \frac{cy}{a} + \frac{az}{b}$ is

The value of $\theta$,satisfying both the equations $\cos \theta = \frac{1}{\sqrt{2}}$ and $\tan \theta = -1$ in the interval $[0, 2\pi]$,is

If $\Delta$ denotes the area of triangle $ABC$,then $(b \sin C + c \sin B)(b \cos C + c \cos B) =$

If the area of a triangle $ABC$ is $\Delta$, then ${a^2}\sin 2B + {b^2}\sin 2A$ is equal to (in $\Delta$)