Proof:

To prove the given equation, we will use the trigonometric identities for the half-angle formulas and the Law of Cosines.

1. Half-angle formulas:
The half-angle formulas for cosine state that:
cos^2(A/2) = (1 + cosA)/2
cos^2(B/2) = (1 + cosB)/2
cos^2(C/2) = (1 + cosC)/2

2. Law of Cosines:
The Law of Cosines states that in a triangle ABC, the square of one side is equal to the sum of the squares of the other two sides minus twice the product of the two sides and the cosine of the included angle. Mathematically, it can be represented as:
c^2 = a^2 + b^2 - 2ab*cosC

Now, let's proceed with the proof:

Step 1: Rewrite the given equation using the half-angle formulas:
4(be cos^2(A/2) ca cos^2(B/2) ab cos^2(C/2)) = (a^2 b^2 c^2)

Step 2: Substitute the half-angle formulas into the equation:
4(be (1 + cosA)/2 ca (1 + cosB)/2 ab (1 + cosC)/2) = (a^2 b^2 c^2)

Step 3: Simplify the equation by multiplying both sides by 2:
2be(1 + cosA)ca(1 + cosB)ab(1 + cosC) = (a^2 b^2 c^2)

Step 4: Expand the equation:
2abc(be + be*cosC + ca + ca*cosB + ab + ab*cosA + be*cosA*cosC + ca*cosB*cosC + ab*cosA*cosB + ab*cosA*cosB*cosC) = (a^2 b^2 c^2)

Step 5: Rearrange the terms and group them:
2abc(be + ca + ab + be*cosC + ca*cosB + ab*cosA + be*cosA*cosC + ca*cosB*cosC + ab*cosA*cosB + ab*cosA*cosB*cosC) = (a^2 b^2 c^2)

Step 6: Apply the Law of Cosines to the grouped terms:
2abc(c + be*cosC + ca*cosB + ab*cosA + ab*cosA*cosB*cosC) = (a^2 b^2 c^2)

Step 7: Simplify the equation further:
2abc(c + be*cosC + ca*cosB + ab*cosA + ab*cosA*cosB*cosC) = (a^2 b^2 c^2)

Step 8: Cancel out common terms:
2(c + be*cosC + ca*cosB + ab*cosA + ab*cosA*cosB*cosC) = (a^2 b^2)

Step 9: Divide both sides by 2:
c + be*cosC + ca*cosB + ab*cosA + ab*cosA*cosB*cosC = (a^2 b^2)/(2c)