Evaluate `(tan60^(@)-tan30^(@))/(1+tan60^(@)tan30^(@))`.

Answer

To evaluate the expression \((\tan 60^\circ - \tan 30^\circ) / (1 + \tan 60^\circ \tan 30^\circ)\), we can follow these steps:

### Step 1: Find the values of \(\tan 60^\circ\) and \(\tan 30^\circ\)

We know from trigonometric ratios:
- \(\tan 60^\circ = \sqrt{3}\)
- \(\tan 30^\circ = \frac{1}{\sqrt{3}}\)

### Step 2: Substitute the values into the expression

Now, substitute these values into the expression:
\[ \frac{\tan 60^\circ - \tan 30^\circ}{1 + \tan 60^\circ \tan 30^\circ} = \frac{\sqrt{3} - \frac{1}{\sqrt{3}}}{1 + \sqrt{3} \cdot \frac{1}{\sqrt{3}}} \]

### Step 3: Simplify the numerator

The numerator becomes:
\[ \sqrt{3} - \frac{1}{\sqrt{3}} = \frac{3}{\sqrt{3}} - \frac{1}{\sqrt{3}} = \frac{3 - 1}{\sqrt{3}} = \frac{2}{\sqrt{3}} \]

### Step 4: Simplify the denominator

The denominator becomes:
\[ 1 + \sqrt{3} \cdot \frac{1}{\sqrt{3}} = 1 + 1 = 2 \]

### Step 5: Combine the results

Now, we can combine the simplified numerator and denominator:
\[ \frac{\frac{2}{\sqrt{3}}}{2} = \frac{2}{\sqrt{3}} \cdot \frac{1}{2} = \frac{1}{\sqrt{3}} \]

### Step 6: Final answer

Thus, the final answer is:
\[ \frac{1}{\sqrt{3}} \]

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