With usual notations in $\triangle ABC$, if $b \cos^2 \frac{C}{2} + c \cos^2 \frac{B}{2} = \frac{3a}{2}$, then:

In a triangle $ABC$,$BC=7$,$AC=8$,$AB=\alpha \in N$ and $\cos A=\frac{2}{3}$. If $49 \cos (3C)+42=\frac{m}{n}$,where $\operatorname{gcd}(m, n)=1$,then $m+n$ is equal to..........

The product of the arithmetic mean of the lengths of the sides of a triangle and the harmonic mean of the lengths of the altitudes of the triangle is equal to: [where $\Delta$ is the area of the triangle $ABC$]

If $ABCD$ is a cyclic quadrilateral,then the value of $\cos A - \cos B + \cos C - \cos D = $

In a triangle $ABC$ with a fixed base $BC$,the vertex $A$ moves such that $\cos B + \cos C = 4 \sin^2 \frac{A}{2}$. If $a, b,$ and $c$ denote the lengths of the sides of the triangle opposite to the angles $A, B,$ and $C$,respectively,then:
$(A) b+c=4a$
$(B) b+c=2a$
$(C) \text{locus of point } A \text{ is an ellipse}$
$(D) \text{locus of point } A \text{ is a pair of straight lines}$

If in a triangle $ABC$,$b \cos^2 \frac{A}{2} + a \cos^2 \frac{B}{2} = \frac{3}{2} c$,then $a, b, c$ are in: