To simplify the expression \(\frac{2 \tan 22.5^\circ}{1 - \tan^2 22.5^\circ}\), we can use the tangent double-angle identity:

\[
\tan(2\theta) = \frac{2 \tan \theta}{1 - \tan^2 \theta}
\]

Here, the given expression matches the formula for \(\tan(2\theta)\). In this case, \(\theta = 22.5^\circ\). 

Therefore:

\[
\frac{2 \tan 22.5^\circ}{1 - \tan^2 22.5^\circ} = \tan(2 \times 22.5^\circ)
\]

\[
\tan(45^\circ) = 1
\]

So, the simplified value is:

\[
\boxed{1}
\]

Would you like more details or have any questions?

Here are 5 related questions to consider:
1. How do you derive the tangent double-angle formula?
2. What is the value of \(\tan(30^\circ)\) using trigonometric identities?
3. Can you simplify \(\frac{1 - \tan^2 \theta}{1 + \tan^2 \theta}\)?
4. What are the identities for \(\sin(2\theta)\) and \(\cos(2\theta)\)?
5. How can you use the tangent double-angle formula to solve trigonometric equations?

**Tip:** Memorizing trigonometric identities can simplify many problems and help you recognize patterns in expressions.

Learn how to simplify the expression 2Tan22.5/1-Tan^222.5 using the tangent double-angle identity in trigonometry. This problem involves applying trigonometric identities to simplify and solve equations. Suitable for high school students and anyone studying trigonometry.

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