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.