Step 1:
First, let's rewrite the right-hand side of the equation using the identity $\cos^2(x) = \frac{1+\cos(2x)}{2}$:
\[2\left(a \cos ^{2} \frac{A}{2}+b \cos ^{2} \frac{B}{2}+c \cos ^{2} \frac{C}{2}\right) = 2\left(a \frac{1+\cos A}{2}+b \frac{1+\cos B}{2}+c
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