Simplify the expression by using a double-angle formula. \[ \frac{2 \tan \frac{\pi}{9}}{1-\tan ^{2} \frac{\pi}{9}} \]

Step 1 ：The given expression is \(\frac{2 \tan \frac{\pi}{9}}{1-\tan ^{2} \frac{\pi}{9}}\)

Step 2 ：This expression is in the form of a double angle formula for tangent, which is given by \(\tan 2 \theta = \frac{2 \tan \theta}{1-\tan ^{2} \theta}\)

Step 3 ：So, we can simplify the given expression by replacing it with the double angle of the angle in the expression

Step 4 ：Let \(\theta = \frac{\pi}{9}\), then the double angle is \(2 \theta = 2 \times \frac{\pi}{9} = \frac{2\pi}{9}\)

Step 5 ：Substitute \(2 \theta\) into the double angle formula, we get \(\tan \frac{2\pi}{9}\)

Step 6 ：Final Answer: The simplified expression is \(\boxed{\tan \frac{2\pi}{9}}\)