Question

If $a{\cos }^2\frac{C}{2}+c{\cos }^2\frac{A}{2}=\frac{3b}{2}$ prove that a, b,c are in A.P.

Answer 1

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

$a\frac{(1+{\cos }^{2}c)}{2}+c\frac{(1+\cos A)}{2}=\frac{3b}{2}$

$a+a\cos c+c+\cos A=3b$

$a+c+a\cos c+\cos A=3b$

$\because a\cos x+\cos A=b$

$a+b+c=3n\Rightarrow a+c=2b$