Question

If in a triangle $a{\cos }^2\frac{C}{2}+c{\cos }^2\frac{A}{2}=\frac{3b}{2}$ then find the relation between the sides of the triangle.

Answer 1

$a{\cos }^2\frac{C}{2}+c{\cos }^2\frac{A}{2}=\frac{3b}{2}$

using trigonometric identitiy
or $a(1+\cos C)+c(1+\cos A)=3b$

or $(a+c)+(a\cos C+c\cos A)=3b$

or $a+c+b=3b$ [by the projection formula]

or $a+c=2b.$ Hence,a,b,c are in A.P.