`(tan 60^(@) - tan 30^(@))/(1 + tan 60^(@) tan 30^(@))` equal

Options

A. `tan 60^(@)`
B. ` tan 0^(@)`
C. `tan 30^(@)`
D. `(1)/(3)`

Answer

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

### Step 1: Identify the values of \(\tan 60^\circ\) and \(\tan 30^\circ\)
We know from trigonometric ratios that:
- \(\tan 60^\circ = \sqrt{3}\)
- \(\tan 30^\circ = \frac{1}{\sqrt{3}}\)

### Step 2: Substitute the values into the expression
Now, we 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 result
Thus, the value of the expression is:
\[ \frac{1}{\sqrt{3}} \]