Let's simplify the given expression step-by-step:
Given expression:
[tex]\[
\frac{\frac{\sqrt{3}}{2} + 1}{1 + \frac{\sqrt{3}}{2}}
\][/tex]
### Step 1: Simplify the Numerator
First, let's simplify the numerator of the expression:
[tex]\[
\frac{\sqrt{3}}{2} + 1
\][/tex]
To add these terms, we can convert 1 into a fraction with the same denominator (which is 2 in this case):
[tex]\[
1 = \frac{2}{2}
\][/tex]
Now, add the terms:
[tex]\[
\frac{\sqrt{3}}{2} + \frac{2}{2} = \frac{\sqrt{3} + 2}{2}
\][/tex]
### Step 2: Simplify the Denominator
Next, let's simplify the denominator of the expression:
[tex]\[
1 + \frac{\sqrt{3}}{2}
\][/tex]
Again, convert 1 into a fraction with the same denominator (which is 2):
[tex]\[
1 = \frac{2}{2}
\][/tex]
Now, add the terms:
[tex]\[
\frac{2}{2} + \frac{\sqrt{3}}{2} = \frac{2 + \sqrt{3}}{2}
\][/tex]
### Step 3: Write the Simplified Expression
Now that we have simplified both the numerator and the denominator, we get:
[tex]\[
\frac{\frac{\sqrt{3} + 2}{2}}{\frac{2 + \sqrt{3}}{2}}
\][/tex]
### Step 4: Simplify the Overall Fraction
To simplify the overall fraction, we can divide the two fractions by multiplying by the reciprocal:
[tex]\[
\frac{\frac{\sqrt{3} + 2}{2}}{\frac{2 + \sqrt{3}}{2}} = \frac{\sqrt{3} + 2}{2} \times \frac{2}{2 + \sqrt{3}}
\][/tex]
Here, the [tex]\(\frac{2}{2}\)[/tex] terms cancel each other out:
[tex]\[
\frac{\sqrt{3} + 2}{2 + \sqrt{3}}
\][/tex]
### Step 5: Simplify the Result
Notice that the numerator and the denominator are now the same:
[tex]\[
\frac{\sqrt{3} + 2}{2 + \sqrt{3}}
\][/tex]
Since they are equivalent, the fraction simplifies to:
[tex]\[
1
\][/tex]
Therefore, the final simplified expression is:
[tex]\[
\boxed{1}
\][/tex]
