Evaluate : `"tan"(2pi)/(9)+"tan"(pi)/(9) + sqrt(3) "tan"(2pi)/(9) "tan"(pi)/(9)`

Answer

To evaluate the expression \( \tan\left(\frac{2\pi}{9}\right) + \tan\left(\frac{\pi}{9}\right) + \sqrt{3} \tan\left(\frac{2\pi}{9}\right) \tan\left(\frac{\pi}{9}\right) \), we can use the tangent addition formula.

### Step-by-step Solution:

1. **Identify the angles**:
Let \( a = \frac{2\pi}{9} \) and \( b = \frac{\pi}{9} \).

2. **Use the tangent addition formula**:
The formula for the tangent of the sum of two angles is:
\[ \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \]
Applying this to our angles:
\[ \tan\left(a + b\right) = \tan\left(\frac{2\pi}{9} + \frac{\pi}{9}\right) = \tan\left(\frac{3\pi}{9}\right) = \tan\left(\frac{\pi}{3}\right) \]

3. **Evaluate \( \tan\left(\frac{\pi}{3}\right) \)**:
We know that:
\[ \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \]

4. **Set up the equation**:
From the tangent addition formula, we have:
\[ \sqrt{3} = \frac{\tan\left(\frac{2\pi}{9}\right) + \tan\left(\frac{\pi}{9}\right)}{1 - \tan\left(\frac{2\pi}{9}\right) \tan\left(\frac{\pi}{9}\right)} \]

5. **Cross-multiply**:
Rearranging gives:
\[ \sqrt{3} \left(1 - \tan\left(\frac{2\pi}{9}\right) \tan\left(\frac{\pi}{9}\right)\right) = \tan\left(\frac{2\pi}{9}\right) + \tan\left(\frac{\pi}{9}\right) \]

6. **Rearranging the equation**:
This can be rewritten as:
\[ \sqrt{3} - \sqrt{3} \tan\left(\frac{2\pi}{9}\right) \tan\left(\frac{\pi}{9}\right) = \tan\left(\frac{2\pi}{9}\right) + \tan\left(\frac{\pi}{9}\right) \]

7. **Combine terms**:
Now, we can rearrange this to isolate the terms:
\[ \tan\left(\frac{2\pi}{9}\right) + \tan\left(\frac{\pi}{9}\right) + \sqrt{3} \tan\left(\frac{2\pi}{9}\right) \tan\left(\frac{\pi}{9}\right) = \sqrt{3} \]

8. **Conclusion**:
Thus, we find that:
\[ \tan\left(\frac{2\pi}{9}\right) + \tan\left(\frac{\pi}{9}\right) + \sqrt{3} \tan\left(\frac{2\pi}{9}\right) \tan\left(\frac{\pi}{9}\right) = \sqrt{3} \]

### Final Answer:
The value of the expression is \( \sqrt{3} \).