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

Options

A. `sin60^(@)`
B. `cos60^(@)`
C. `tan60^(@)`
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 the value of \(\tan 30^\circ\)
We know that:
\[ \tan 30^\circ = \frac{1}{\sqrt{3}} \]

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

### Step 5: Simplify the expression
Now simplify the expression:
\[ = \frac{2 \cdot \frac{1}{\sqrt{3}} \cdot 3}{4} = \frac{6}{4\sqrt{3}} = \frac{3}{2\sqrt{3}} \]

### Step 6: Rationalize the denominator
To rationalize the denominator:
\[ \frac{3}{2\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{3\sqrt{3}}{6} = \frac{\sqrt{3}}{2} \]

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