In a $\triangle ABC$,if $3a = b + c$,then $\cot \frac{B}{2} \cot \frac{C}{2}$ is equal to :

If the angles of a triangle $ABC$ are in the ratio $1: 2: 3$,then the corresponding sides are in the ratio

The sides of a triangle are $\sin \alpha$,$\cos \alpha$,and $\sqrt{1 + \sin \alpha \cos \alpha}$ for some $0 < \alpha < \frac{\pi}{2}$. Then the greatest angle of the triangle is.....$^o$

In a triangle $ABC$ with usual notations,if $\tan \left(\frac{B-C}{2}\right) = x \cot \frac{A}{2}$,then $x =$

If $ABC$ is a right-angled triangle with $90^{\circ}$ at $C$ and $a > b$,then $\frac{a^2+b^2}{a^2-b^2} \sin (A-B) = $

In a triangle $ABC$ with usual notations $a=2$ and $b=3$,the value of $\frac{\cos 2A}{a^2} - \frac{\cos 2B}{b^2}$ is: