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In a $\triangle A B C$, if $a \cos ^2 \frac{C}{2}+c \cos ^2 \frac{A}{2}=\frac{3 b}{2}$, then
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The correct answer is:
$2 b=a+c$
$\begin{aligned} & \text { Given, } a \cos ^2 \frac{C}{2}+c \cos ^2 \frac{A}{2}=\frac{3 b}{2} \\ & \Rightarrow a(1+\cos c)+c(1+\cos A)=3 b \\ & \Rightarrow a+c+(a \cos C+c \cos A)=3 b \\ & \Rightarrow(a+c)+b=3 b \Rightarrow a+c=2 b\end{aligned}$
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