To find the quotient of [tex]\(\frac{1}{1 + \sqrt{3}}\)[/tex] and express it in a simplified form, we need to rationalize the denominator. Here is the step-by-step solution:
1. Identify the expression:
[tex]\[
\frac{1}{1 + \sqrt{3}}
\][/tex]
2. Multiply by the conjugate of the denominator:
The conjugate of [tex]\(1 + \sqrt{3}\)[/tex] is [tex]\(1 - \sqrt{3}\)[/tex]. Thus, we multiply both numerator and denominator by this conjugate to remove the square root from the denominator:
[tex]\[
\frac{1}{1 + \sqrt{3}} \cdot \frac{1 - \sqrt{3}}{1 - \sqrt{3}} = \frac{1 \cdot (1 - \sqrt{3})}{(1 + \sqrt{3}) \cdot (1 - \sqrt{3})}
\][/tex]
3. Calculate the new numerator:
[tex]\[
1 \cdot (1 - \sqrt{3}) = 1 - \sqrt{3}
\][/tex]
4. Calculate the new denominator:
The denominator is a difference of squares:
[tex]\[
(1 + \sqrt{3}) \cdot (1 - \sqrt{3}) = 1^2 - (\sqrt{3})^2 = 1 - 3 = -2
\][/tex]
5. Form the rationalized fraction:
[tex]\[
\frac{1 - \sqrt{3}}{-2}
\][/tex]
6. Simplify the fraction:
Dividing each term in the numerator by the denominator:
[tex]\[
\frac{1 - \sqrt{3}}{-2} = -\frac{1}{2} + \frac{\sqrt{3}}{2}
\][/tex]
7. Match the form with one of the given choices:
The simplified form is equivalent to:
[tex]\[
-\frac{1}{2} + \frac{\sqrt{3}}{2} \equiv \frac{-1 + \sqrt{3}}{2}
\][/tex]
Therefore, the correct quotient is:
[tex]\[
\boxed{\frac{-1 + \sqrt{3}}{2}}
\][/tex]
