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The expression you provided is:

\[
\frac{2 \tan 30^\circ}{1 + \tan^2 30^\circ}
\]

Let's calculate this step-by-step.

### Step 1: Find \(\tan 30^\circ\)

\[
\tan 30^\circ = \frac{1}{\sqrt{3}}
\]

### Step 2: Substitute \(\tan 30^\circ\) into the expression

\[
\frac{2 \cdot \frac{1}{\sqrt{3}}}{1 + \left( \frac{1}{\sqrt{3}} \right)^2}
\]

### Step 3: Simplify the denominator

\[
1 + \left( \frac{1}{\sqrt{3}} \right)^2 = 1 + \frac{1}{3} = \frac{4}{3}
\]

### Step 4: Substitute back into the expression

\[
\frac{\frac{2}{\sqrt{3}}}{\frac{4}{3}} = \frac{2}{\sqrt{3}} \cdot \frac{3}{4} = \frac{6}{4\sqrt{3}} = \frac{3}{2\sqrt{3}}
\]

### Step 5: Rationalize the denominator

\[
\frac{3}{2\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{3\sqrt{3}}{2 \cdot 3} = \frac{\sqrt{3}}{2}
\]

### Final Answer

\[
\frac{2 \tan 30^\circ}{1 + \tan^2 30^\circ} = \frac{\sqrt{3}}{2}
\]

Would you like to explore this further or ask any questions? Here are some related questions:

1. How can you find the values of other trigonometric functions at different angles?
2. What are the properties of trigonometric identities used here?
3. How do you rationalize a denominator in general?
4. Can this formula be generalized for other angles?
5. What are some practical applications of trigonometric ratios?

**Tip:** Understanding the fundamental trigonometric values like \(\tan 30^\circ\), \(\sin 45^\circ\), etc., will make solving these types of problems much easier.

Find the value of (2 * tan 30°) / (1 + tan² 30°).

Learn how to evaluate the expression (2 * tan 30°) / (1 + tan² 30°) using basic trigonometric ratios and simplifying the result. This step-by-step solution involves simplifying square roots and rationalizing the denominator. Ideal for high school students (Grades 10-12).

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