https://file.aiquickdraw.com/mathbot/file/1725538181674geg2bq37.jpg

The expression given in the image is:

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

Let's break this down and solve it step-by-step.

1. **Step 1: Evaluate \(\tan 30^\circ\)**

The value of \(\tan 30^\circ\) is well-known and equals:

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

2. **Step 2: Substitute into the expression**

Now, substitute \(\tan 30^\circ = \frac{1}{\sqrt{3}}\) into the given equation:

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

3. **Step 3: Simplify the denominator**

First, simplify the square of \(\frac{1}{\sqrt{3}}\):

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

So, the denominator becomes:

\[
1 + \frac{1}{3} = \frac{4}{3}
\]

4. **Step 4: Simplify the whole expression**

Now the expression becomes:

\[
\frac{2 \cdot \frac{1}{\sqrt{3}}}{\frac{4}{3}} = 2 \cdot \frac{1}{\sqrt{3}} \cdot \frac{3}{4}
\]

Simplify the multiplication:

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

5. **Step 5: Rationalize the denominator**

To rationalize the denominator, multiply both the numerator and denominator by \(\sqrt{3}\):

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

Thus, the simplified result is:

\[
\boxed{\frac{\sqrt{3}}{2}}
\]

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

1. How do you compute other trigonometric identities like \(\sin 30^\circ\) and \(\cos 30^\circ\)?
2. How do you rationalize a denominator in general?
3. What are other methods for simplifying trigonometric expressions?
4. How does the tangent function behave for other common angles like \(45^\circ\) and \(60^\circ\)?
5. Can you solve the same expression for different angles, like \(45^\circ\) or \(60^\circ\)?

**Tip:** Remember, it's useful to memorize key trigonometric values for angles like \(30^\circ\), \(45^\circ\), and \(60^\circ\), as they often appear in problems like this one.

Find \( \frac{2 \cdot \tan 30^\circ}{1 + \tan^2 30^\circ} \)

Learn how to simplify the trigonometric expression \( \frac{2 \cdot \tan 30^\circ}{1 + \tan^2 30^\circ} \). This detailed solution involves key trigonometric concepts, including tangent values for common angles and rationalizing square roots. Suitable for high school students in Grades 10-12.

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