In $\triangle ABC$,if $x=\tan \left(\frac{B-C}{2}\right) \tan \frac{A}{2}$,$y=\tan \left(\frac{C-A}{2}\right) \tan \frac{B}{2}$,and $z=\tan \left(\frac{A-B}{2}\right) \tan \frac{C}{2}$,then $(x+y+z)$ is equal to

In a $\triangle PQR$,$m \angle R = \frac{\pi}{2}$. If $\tan \left(\frac{P}{2}\right)$ and $\tan \left(\frac{Q}{2}\right)$ are the roots of the equation $ax^2 + bx + c = 0$ $(a \neq 0)$,then:

In a $\triangle ABC$,the value of $\Sigma(b+c) \tan \frac{A}{2} \tan \left(\frac{B-C}{2}\right)$ is equal to

Consider the following statements.
$I$. In $\triangle ABC$,if $c=6$ and $\cos C=-\frac{11}{25}$,then $R=\frac{25}{2\sqrt{14}}$.
$II$. In $\triangle ABC$,if $a=3, b=4, c=6$,then $\triangle ABC$ is an acute-angled triangle.
Which of the above statements is/are true?

If in a triangle the angles are in $A.P.$ and $b:c = \sqrt{3}:\sqrt{2}$,then $\angle A$ is equal to .....$^o$.

In a triangle,if the lengths of the sides $a, b,$ and $c$ are three consecutive natural numbers and $a < b < c$,then $(\cos A + \cos B + \cos C) 2abc = $