The value of `[ ( sin ""(pi)/(9)) (4+ sec""(pi)/(9))]^(2)` is _______.

Explanation

To solve the problem, we need to find the value of \(\left( \sin\left(\frac{\pi}{9}\right) \left(4 + \sec\left(\frac{\pi}{9}\right)\right)\right)^2\).

### Step-by-Step Solution:

1. **Rewrite the secant function**:
\[ \sec\left(\frac{\pi}{9}\right) = \frac{1}{\cos\left(\frac{\pi}{9}\right)} \]
Therefore, we can rewrite the expression as:
\[ \sin\left(\frac{\pi}{9}\right) \left(4 + \sec\left(\frac{\pi}{9}\right)\right) = \sin\left(\frac{\pi}{9}\right) \left(4 + \frac{1}{\cos\left(\frac{\pi}{9}\right)}\right) \]

2. **Combine the terms**:
\[ = \sin\left(\frac{\pi}{9}\right) \left(\frac{4 \cos\left(\frac{\pi}{9}\right) + 1}{\cos\left(\frac{\pi}{9}\right)}\right) \]
This simplifies to:
\[ = \frac{\sin\left(\frac{\pi}{9}\right) (4 \cos\left(\frac{\pi}{9}\right) + 1)}{\cos\left(\frac{\pi}{9}\right)} \]

3. **Distribute \(\sin\left(\frac{\pi}{9}\right)\)**:
\[ = \frac{4 \sin\left(\frac{\pi}{9}\right) \cos\left(\frac{\pi}{9}\right) + \sin\left(\frac{\pi}{9}\right)}{\cos\left(\frac{\pi}{9}\right)} \]

4. **Use the double angle identity**:
Recall that \(2 \sin\theta \cos\theta = \sin(2\theta)\):
\[ = \frac{2 \sin\left(\frac{2\pi}{9}\right) + \sin\left(\frac{\pi}{9}\right)}{\cos\left(\frac{\pi}{9}\right)} \]

5. **Combine the sine terms**:
Using the sine addition formula:
\[ \sin A + \sin B = 2 \sin\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right) \]
Let \(A = \frac{2\pi}{9}\) and \(B = \frac{\pi}{9}\):
\[ = \frac{2 \sin\left(\frac{3\pi}{9}\right) \cos\left(\frac{\pi}{9}\right)}{\cos\left(\frac{\pi}{9}\right)} = 2 \sin\left(\frac{\pi}{3}\right) \]

6. **Evaluate \(\sin\left(\frac{\pi}{3}\right)\)**:
\[ \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \]
Thus:
\[ = 2 \cdot \frac{\sqrt{3}}{2} = \sqrt{3} \]

7. **Square the result**:
\[ \left(\sqrt{3}\right)^2 = 3 \]

### Final Answer:
The value of \(\left( \sin\left(\frac{\pi}{9}\right) \left(4 + \sec\left(\frac{\pi}{9}\right)\right)\right)^2\) is \(\boxed{3}\).