Prove that `cos^3A+cos^3(120^0+A)+cos^3(240^0+A)=3/4cos3A`

Explanation

To prove that \( \cos^3 A + \cos^3(120^\circ + A) + \cos^3(240^\circ + A) = \frac{3}{4} \cos 3A \), we will follow these steps:

### Step 1: Use the identity for the sum of cubes
We start with the expression \( \cos^3 A + \cos^3(120^\circ + A) + \cos^3(240^\circ + A) \). We can use the identity for the sum of cubes:
\[ x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - xz - yz) \]
In our case, let \( x = \cos A \), \( y = \cos(120^\circ + A) \), and \( z = \cos(240^\circ + A) \).
...