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: Identify 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, substituting 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
To simplify the numerator, we need a common denominator:
\[ \sqrt{3} - \frac{1}{\sqrt{3}} = \frac{3}{\sqrt{3}} - \frac{1}{\sqrt{3}} = \frac{3 - 1}{\sqrt{3}} = \frac{2}{\sqrt{3}} \]

### Step 4: Simplify the denominator
Now, simplifying the denominator:
\[ 1 + \sqrt{3} \cdot \frac{1}{\sqrt{3}} = 1 + 1 = 2 \]

### Step 5: Combine the results
Now we can substitute back into the expression:
\[ \frac{\frac{2}{\sqrt{3}}}{2} \]

### Step 6: Simplify the fraction
This simplifies to:
\[ \frac{2}{\sqrt{3}} \cdot \frac{1}{2} = \frac{1}{\sqrt{3}} \]

### Final Answer
Thus, the value of the expression is:
\[ \frac{1}{\sqrt{3}} \]

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