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In any triangle A B C , prove that: a^3c...

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

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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. ...
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