Value of `(2tan30^(@))/(1+ tan^(2)30^(@))`

Options

A. `sin 60^(@)`
B. `cos 60^(@)`
C. `tan60^(@)`
D. ` sin 30^(@)`

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 \(\tan 30^\circ\)
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: Simplify the denominator
Calculate \(\tan^2 30^\circ\):
\[ \tan^2 30^\circ = \left(\frac{1}{\sqrt{3}}\right)^2 = \frac{1}{3} \]
Now substitute this back into the denominator:
\[ 1 + \tan^2 30^\circ = 1 + \frac{1}{3} = \frac{3}{3} + \frac{1}{3} = \frac{4}{3} \]

### Step 4: Substitute back into the expression
Now we have:
\[ \frac{2 \cdot \frac{1}{\sqrt{3}}}{\frac{4}{3}} = \frac{\frac{2}{\sqrt{3}}}{\frac{4}{3}} \]

### Step 5: Simplify the fraction
To simplify \(\frac{\frac{2}{\sqrt{3}}}{\frac{4}{3}}\), we multiply by the reciprocal:
\[ \frac{2}{\sqrt{3}} \cdot \frac{3}{4} = \frac{2 \cdot 3}{\sqrt{3} \cdot 4} = \frac{6}{4\sqrt{3}} = \frac{3}{2\sqrt{3}} \]

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

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