The value of `(2tan30^(@))/(1-tan^(2)30^(@))` is :

Options

A. `tan60^(@)`
B. `sin60^(@)`
C. `cos60^(@)`
D. `cot60^(@)`

Answer

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

### Step 1: Find \(\tan 30^\circ\)
We know that:
\[ \tan 30^\circ = \frac{1}{\sqrt{3}} \]

### 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 \(\tan^2 30^\circ\) 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 can rewrite the expression:
\[ \frac{2 \cdot \frac{1}{\sqrt{3}}}{\frac{2}{3}} = \frac{2 \cdot \frac{1}{\sqrt{3}} \cdot 3}{2} \]

### Step 7: Cancel out the common terms
The 2 in the numerator and denominator cancels out:
\[ = \frac{3}{\sqrt{3}} \]

### Step 8: Simplify further
Now we simplify \(\frac{3}{\sqrt{3}}\):
\[ \frac{3}{\sqrt{3}} = \sqrt{3} \]

### Conclusion
Thus, the value of \(\frac{2\tan 30^\circ}{1 - \tan^2 30^\circ}\) is:
\[ \sqrt{3} \]