Evaluate :
`8 cos ^(3) ""(pi)/(9)- 6cos ""(pi)/(9)`

Answer

To evaluate the expression \( 8 \cos^3 \left( \frac{\pi}{9} \right) - 6 \cos \left( \frac{\pi}{9} \right) \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ 8 \cos^3 \left( \frac{\pi}{9} \right) - 6 \cos \left( \frac{\pi}{9} \right) \]
We can factor out a common factor of \( 2 \):
\[ = 2 \left( 4 \cos^3 \left( \frac{\pi}{9} \right) - 3 \cos \left( \frac{\pi}{9} \right) \right) \]

### Step 2: Use the cosine triple angle formula
We know from trigonometric identities that:
\[ \cos(3\theta) = 4 \cos^3(\theta) - 3 \cos(\theta) \]
In our case, let \( \theta = \frac{\pi}{9} \). Therefore, we can rewrite the expression inside the parentheses:
\[ 4 \cos^3 \left( \frac{\pi}{9} \right) - 3 \cos \left( \frac{\pi}{9} \right) = \cos(3 \cdot \frac{\pi}{9}) = \cos \left( \frac{\pi}{3} \right) \]

### Step 3: Evaluate \( \cos \left( \frac{\pi}{3} \right) \)
We know that:
\[ \cos \left( \frac{\pi}{3} \right) = \frac{1}{2} \]

### Step 4: Substitute back into the expression
Now we can substitute this value back into our expression:
\[ 2 \left( \cos \left( \frac{\pi}{3} \right) \right) = 2 \left( \frac{1}{2} \right) = 1 \]

### Final Answer
Thus, the value of the original expression \( 8 \cos^3 \left( \frac{\pi}{9} \right) - 6 \cos \left( \frac{\pi}{9} \right) \) is:
\[ \boxed{1} \]

---