`((2+sqrt(3))/(2-sqrt(3))+(2-sqrt(3))/(2+sqrt(3))+(sqrt(3)-1)/(sqrt(3)+1))` simplifies to :

Options

A. `2-sqrt(3)`
B. `2+sqrt(3)`
C. `16-sqrt(3)`
D. `40-sqrt(3)`

Answer

To simplify the expression \(\frac{(2+\sqrt{3})}{(2-\sqrt{3})} + \frac{(2-\sqrt{3})}{(2+\sqrt{3})} + \frac{(\sqrt{3}-1)}{(\sqrt{3}+1)}\), we will follow these steps:

### Step 1: Simplify the first two fractions
We start with the first two fractions:

\[ \frac{(2+\sqrt{3})}{(2-\sqrt{3})} + \frac{(2-\sqrt{3})}{(2+\sqrt{3})} \]

To simplify these fractions, we will rationalize them. We can multiply the first fraction by \(\frac{(2+\sqrt{3})}{(2+\sqrt{3})}\) and the second fraction by \(\frac{(2-\sqrt{3})}{(2-\sqrt{3})}\).

### Step 2: Rationalizing the first fraction
For the first fraction:

\[ \frac{(2+\sqrt{3})}{(2-\sqrt{3})} \cdot \frac{(2+\sqrt{3})}{(2+\sqrt{3})} = \frac{(2+\sqrt{3})^2}{(2-\sqrt{3})(2+\sqrt{3})} \]

Calculating the denominator:

\[ (2-\sqrt{3})(2+\sqrt{3}) = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1 \]

Calculating the numerator:

\[ (2+\sqrt{3})^2 = 2^2 + 2 \cdot 2 \cdot \sqrt{3} + (\sqrt{3})^2 = 4 + 4\sqrt{3} + 3 = 7 + 4\sqrt{3} \]

Thus, the first fraction simplifies to:

\[ \frac{7 + 4\sqrt{3}}{1} = 7 + 4\sqrt{3} \]

### Step 3: Rationalizing the second fraction
Now for the second fraction:

\[ \frac{(2-\sqrt{3})}{(2+\sqrt{3})} \cdot \frac{(2-\sqrt{3})}{(2-\sqrt{3})} = \frac{(2-\sqrt{3})^2}{(2+\sqrt{3})(2-\sqrt{3})} \]

The denominator is already calculated as 1. Now calculating the numerator:

\[ (2-\sqrt{3})^2 = 2^2 - 2 \cdot 2 \cdot \sqrt{3} + (\sqrt{3})^2 = 4 - 4\sqrt{3} + 3 = 7 - 4\sqrt{3} \]

Thus, the second fraction simplifies to:

\[ \frac{7 - 4\sqrt{3}}{1} = 7 - 4\sqrt{3} \]

### Step 4: Adding the first two fractions
Now we add the two simplified fractions:

\[ (7 + 4\sqrt{3}) + (7 - 4\sqrt{3}) = 7 + 7 + 4\sqrt{3} - 4\sqrt{3} = 14 \]

### Step 5: Simplifying the third fraction
Now we simplify the third fraction:

\[ \frac{(\sqrt{3}-1)}{(\sqrt{3}+1)} \]

We rationalize this by multiplying by \(\frac{(\sqrt{3}-1)}{(\sqrt{3}-1)}\):

\[ \frac{(\sqrt{3}-1)^2}{(\sqrt{3}+1)(\sqrt{3}-1)} \]

Calculating the denominator:

\[ (\sqrt{3}+1)(\sqrt{3}-1) = (\sqrt{3})^2 - 1^2 = 3 - 1 = 2 \]

Calculating the numerator:

\[ (\sqrt{3}-1)^2 = (\sqrt{3})^2 - 2 \cdot \sqrt{3} \cdot 1 + 1^2 = 3 - 2\sqrt{3} + 1 = 4 - 2\sqrt{3} \]

Thus, the third fraction simplifies to:

\[ \frac{4 - 2\sqrt{3}}{2} = 2 - \sqrt{3} \]

### Step 6: Adding all parts together
Now we add the result from Step 4 and Step 5:

\[ 14 + (2 - \sqrt{3}) = 16 - \sqrt{3} \]

### Final Answer
Thus, the expression simplifies to:

\[ \boxed{16 - \sqrt{3}} \]