Given:
A(CosC - CosB) = 2(b-c) cos^2 (A/2)

To Prove:
A(CosC - CosB) = 2(b-c) cos^2 (A/2)

Proof:

Step 1: Simplify the right side of the equation

Using the identity cos^2 (A/2) = (1 + cos A)/2, we can rewrite the equation as:

A(CosC - CosB) = 2(b-c) (1 + cos A)/2

Simplifying further, we have:

A(CosC - CosB) = (b - c)(1 + cos A)

Step 2: Expand the equation

Expanding the equation, we get:

ACosC - ACosB = b - c + bcosA - ccosA

Step 3: Rearrange the terms

Rearranging the terms, we have:

ACosC - bcosA = ACosB - ccosA + b - c

Step 4: Use the Law of Cosines

Using the Law of Cosines, we know that:

b^2 = a^2 + c^2 - 2acCosB

Rearranging this equation, we get:

2acCosB = a^2 + c^2 - b^2

Similarly:

2bcCosA = b^2 + c^2 - a^2

Substituting these equations into our rearranged equation from Step 3, we have:

ACosC - bcosA = a^2 + c^2 - b^2 - ccosA + b - c

Step 5: Simplify the equation

Simplifying the equation, we get:

ACosC - bcosA = a^2 - b^2 + c^2 - ccosA + b - c

Step 6: Rearrange the terms

Rearranging the terms, we have:

ACosC + ccosA = a^2 - b^2 + c^2 + b - c

Step 7: Use the Law of Cosines once again

Using the Law of Cosines, we know that:

a^2 = b^2 + c^2 - 2bccosA

Substituting this equation into our rearranged equation from Step 6, we have:

ACosC + ccosA = b^2 + c^2 - 2bccosA - b^2 + c^2 + b - c

Step 8: Simplify the equation

Simplifying the equation, we get:

ACosC + ccosA = 2c^2 - 2bccosA + b - c

Step 9: Rearrange the terms

Rearranging the terms, we have:

-2bccosA - ccosA + ACosC + c = b - c + 2c^2

Step 10: Use the Law of Cosines once again

Using the Law of Cosines, we know that:

c^2 = a^2