If `x=sqrt(1+sqrt(3)/(2))-sqrt(1-sqrt(3)/(2))`, then the value of `(sqrt(2)-x)/(sqrt(2)+x)` will be closet to :

Options

A. 1.4
B. 1.2
C. 0.17
D. 0.12

Answer

To solve the problem step by step, we start with the given expression for \( x \):

\[ x = \sqrt{1 + \frac{\sqrt{3}}{2}} - \sqrt{1 - \frac{\sqrt{3}}{2}} \]

### Step 1: Simplify \( x \)

First, we simplify the terms inside the square roots.

1. Calculate \( 1 + \frac{\sqrt{3}}{2} \):
\[ 1 + \frac{\sqrt{3}}{2} = \frac{2}{2} + \frac{\sqrt{3}}{2} = \frac{2 + \sqrt{3}}{2} \]

2. Calculate \( 1 - \frac{\sqrt{3}}{2} \):
\[ 1 - \frac{\sqrt{3}}{2} = \frac{2}{2} - \frac{\sqrt{3}}{2} = \frac{2 - \sqrt{3}}{2} \]

Now, substituting these back into the expression for \( x \):
\[ x = \sqrt{\frac{2 + \sqrt{3}}{2}} - \sqrt{\frac{2 - \sqrt{3}}{2}} \]

### Step 2: Factor out the square root of \( \frac{1}{2} \)

We can factor out \( \sqrt{\frac{1}{2}} \) from both terms:
\[ x = \sqrt{\frac{1}{2}} \left( \sqrt{2 + \sqrt{3}} - \sqrt{2 - \sqrt{3}} \right) \]

### Step 3: Calculate \( \sqrt{2 + \sqrt{3}} \) and \( \sqrt{2 - \sqrt{3}} \)

To find \( \sqrt{2 + \sqrt{3}} \) and \( \sqrt{2 - \sqrt{3}} \), we can approximate their values:

1. \( \sqrt{2 + \sqrt{3}} \approx \sqrt{2 + 1.732} \approx \sqrt{3.732} \approx 1.93 \)
2. \( \sqrt{2 - \sqrt{3}} \approx \sqrt{2 - 1.732} \approx \sqrt{0.268} \approx 0.52 \)

### Step 4: Substitute back to find \( x \)

Now substituting these approximations back into the expression for \( x \):
\[ x \approx \sqrt{\frac{1}{2}} (1.93 - 0.52) = \sqrt{\frac{1}{2}} (1.41) \approx 1.41 \times 0.707 \approx 1.00 \]

### Step 5: Calculate \( \frac{\sqrt{2} - x}{\sqrt{2} + x} \)

Now we need to find the value of:
\[ \frac{\sqrt{2} - x}{\sqrt{2} + x} \]

Using \( \sqrt{2} \approx 1.41 \) and \( x \approx 1.00 \):
\[ \frac{1.41 - 1.00}{1.41 + 1.00} = \frac{0.41}{2.41} \approx 0.17 \]

### Final Answer

The value of \( \frac{\sqrt{2} - x}{\sqrt{2} + x} \) is approximately \( 0.17 \).