Find the vlaue of `( tan 60^(@) - tan 15 ^(@))/(1 + tan 60^(@)tan 15 ^(@))`

Options

A. A) 1
B. B) `(1)/(sqrt2)`
C. C) `(1)/(2)`
D. D) `(sqrt3)/(2)`

Answer

To solve the expression \(( \tan 60^\circ - \tan 15^\circ ) / (1 + \tan 60^\circ \tan 15^\circ)\), we can use the tangent subtraction formula.

### Step-by-Step Solution:

1. **Identify the angles**: We have \(a = 60^\circ\) and \(b = 15^\circ\).

2. **Use the tangent subtraction formula**: The formula for \(\tan(a - b)\) is given by:
\[ \tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b} \]
Here, we can see that our expression matches this formula.

3. **Calculate \(a - b\)**:
\[ a - b = 60^\circ - 15^\circ = 45^\circ \]

4. **Evaluate \(\tan 45^\circ\)**:
\[ \tan 45^\circ = 1 \]

5. **Conclusion**: Therefore, the value of the expression is:
\[ \frac{\tan 60^\circ - \tan 15^\circ}{1 + \tan 60^\circ \tan 15^\circ} = \tan(45^\circ) = 1 \]

### Final Answer:
The value of \(( \tan 60^\circ - \tan 15^\circ ) / (1 + \tan 60^\circ \tan 15^\circ)\) is \(1\).

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