Value of `(2 tan^(2) 60^(@))/(1 + tan^(2) 30^(@))` = _________.

Answer

To solve the expression \(\frac{2 \tan^2 60^\circ}{1 + \tan^2 30^\circ}\), we will follow these steps:

### Step 1: Find the values of \(\tan^2 60^\circ\) and \(\tan^2 30^\circ\)

- We know that:
\[ \tan 60^\circ = \sqrt{3} \]
Therefore,
\[ \tan^2 60^\circ = (\sqrt{3})^2 = 3 \]

- Also, we know that:
\[ \tan 30^\circ = \frac{1}{\sqrt{3}} \]
Therefore,
\[ \tan^2 30^\circ = \left(\frac{1}{\sqrt{3}}\right)^2 = \frac{1}{3} \]

### Step 2: Substitute these values into the expression

Now substituting the values we found into the expression:
\[ \frac{2 \tan^2 60^\circ}{1 + \tan^2 30^\circ} = \frac{2 \cdot 3}{1 + \frac{1}{3}} \]

### Step 3: Simplify the denominator

Now, simplify the denominator:
\[ 1 + \frac{1}{3} = \frac{3}{3} + \frac{1}{3} = \frac{4}{3} \]

### Step 4: Rewrite the expression

Now we can rewrite the expression:
\[ \frac{2 \cdot 3}{\frac{4}{3}} = \frac{6}{\frac{4}{3}} \]

### Step 5: Simplify the fraction

To simplify \(\frac{6}{\frac{4}{3}}\), we multiply by the reciprocal:
\[ 6 \cdot \frac{3}{4} = \frac{18}{4} \]

### Step 6: Reduce the fraction

Now, reduce \(\frac{18}{4}\):
\[ \frac{18}{4} = \frac{9}{2} \]

### Final Answer

Thus, the value of the expression \(\frac{2 \tan^2 60^\circ}{1 + \tan^2 30^\circ}\) is:
\[ \frac{9}{2} \]

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