`(2tan60^(@))/(1+tan^(2)60^(@))` is equal to :

Options

A. `sin60^(@)`
B. `cos60^(@)`
C. `tan30^(@)`
D. `sin 30^(@)`

Answer

To solve the expression \((2\tan 60^\circ)/(1+\tan^2 60^\circ)\), we can follow these steps:

### Step 1: Find \(\tan 60^\circ\)
We know that:
\[ \tan 60^\circ = \sqrt{3} \]

### Step 2: Substitute \(\tan 60^\circ\) into the expression
Now, we substitute \(\tan 60^\circ\) into the expression:
\[ \frac{2\tan 60^\circ}{1+\tan^2 60^\circ} = \frac{2\sqrt{3}}{1+\tan^2 60^\circ} \]

### Step 3: Calculate \(\tan^2 60^\circ\)
Next, we calculate \(\tan^2 60^\circ\):
\[ \tan^2 60^\circ = (\sqrt{3})^2 = 3 \]

### Step 4: Substitute \(\tan^2 60^\circ\) into the expression
Now, substitute \(\tan^2 60^\circ\) back into the expression:
\[ \frac{2\sqrt{3}}{1+3} = \frac{2\sqrt{3}}{4} \]

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

### Conclusion
Thus, the value of \(\frac{2\tan 60^\circ}{1+\tan^2 60^\circ}\) is:
\[ \frac{\sqrt{3}}{2} \]