Find the vlaue of `(2tan30^(@))/(1-tan^(2) 30^(@))`.

Answer

To find the value of the expression \(\frac{2 \tan 30^\circ}{1 - \tan^2 30^\circ}\), we can follow these steps:

### Step 1: Find \(\tan 30^\circ\)
The value of \(\tan 30^\circ\) is known to be \(\frac{1}{\sqrt{3}}\).

### Step 2: Substitute \(\tan 30^\circ\) into the expression
Now, we substitute \(\tan 30^\circ\) into the expression:
\[ \frac{2 \tan 30^\circ}{1 - \tan^2 30^\circ} = \frac{2 \cdot \frac{1}{\sqrt{3}}}{1 - \left(\frac{1}{\sqrt{3}}\right)^2} \]

### Step 3: Calculate \(\tan^2 30^\circ\)
Next, we calculate \(\tan^2 30^\circ\):
\[ \tan^2 30^\circ = \left(\frac{1}{\sqrt{3}}\right)^2 = \frac{1}{3} \]

### Step 4: Substitute \(\tan^2 30^\circ\) into the expression
Now we substitute this value back into the expression:
\[ \frac{2 \cdot \frac{1}{\sqrt{3}}}{1 - \frac{1}{3}} = \frac{2 \cdot \frac{1}{\sqrt{3}}}{\frac{2}{3}} \]

### Step 5: Simplify the denominator
The denominator simplifies as follows:
\[ 1 - \frac{1}{3} = \frac{2}{3} \]

### Step 6: Rewrite the expression
Now we rewrite the expression:
\[ \frac{2 \cdot \frac{1}{\sqrt{3}}}{\frac{2}{3}} = \frac{2}{\sqrt{3}} \cdot \frac{3}{2} \]

### Step 7: Cancel the common terms
We can cancel the \(2\) in the numerator and denominator:
\[ \frac{2}{\sqrt{3}} \cdot \frac{3}{2} = \frac{3}{\sqrt{3}} \]

### Step 8: Simplify the final expression
Now, we simplify \(\frac{3}{\sqrt{3}}\):
\[ \frac{3}{\sqrt{3}} = \sqrt{3} \]

### Final Answer
Thus, the value of \(\frac{2 \tan 30^\circ}{1 - \tan^2 30^\circ}\) is \(\sqrt{3}\).

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