In any triangle `A B C ,` prove that: `a^3cos(B-C)+b^3cos(C-A)+c^3cos(A-B)=3a b c`

Explanation

To prove the equation \( a^3 \cos(B - C) + b^3 \cos(C - A) + c^3 \cos(A - B) = 3abc \) for any triangle \( ABC \), we can follow these steps:

### Step 1: Use the Cosine Rule
We start with the left-hand side (LHS):
\[ LHS = a^3 \cos(B - C) + b^3 \cos(C - A) + c^3 \cos(A - B) \]
Using the cosine of the difference formula, we can express \( \cos(B - C) \), \( \cos(C - A) \), and \( \cos(A - B) \) in terms of the sides and angles of the triangle.
...