For a $\Delta ABC$,if $a \cos^2 \frac{C}{2} + c \cos^2 \frac{A}{2} = \frac{3b}{2}$,then the sides $a, b, c$ are in:

Which of the following is true?

If in a triangle $ABC$,$a^2+2bc-(b^2+c^2)=ab \sin \frac{C}{2} \cos \frac{C}{2}$,then $\cot (B+C)=$

Let the angles $A, B, C$ of a triangle $ABC$ be in arithmetic progression. If the exradii $r_1, r_2, r_3$ of triangle $ABC$ satisfy the condition $r_3^2 = r_1 r_2 + r_2 r_3 + r_3 r_1$, then $b =$

In a $\triangle ABC$,$\cos \left(\frac{B+2C+3A}{2}\right) + \cos \left(\frac{A-B}{2}\right)$ is equal to

Assertion $(A)$: If $A=15^{\circ}, B=17^{\circ}$ and $C=13^{\circ}$,then $\cot 2A + \cot 2B + \cot 2C = \cot 2A \cot 2B \cot 2C$.
Reason $(R)$: In a $\triangle PQR$,$\tan \frac{P}{2} \tan \frac{Q}{2} + \tan \frac{Q}{2} \tan \frac{R}{2} + \tan \frac{P}{2} \tan \frac{R}{2} = 1$.
The correct option among the following is: