On simplification the value of `1-(1)/(1+sqrt(2))+(1)/(1-sqrt(2))` is

Options

A. `2sqrt(2)-1`
B. `1-2sqrt(2)`
C. `1-sqrt(2)`
D. `-2sqrt(2)`

Answer

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

### Step 1: Simplify \( \frac{1}{1+\sqrt{2}} \)
To simplify \( \frac{1}{1+\sqrt{2}} \), we multiply the numerator and denominator by the conjugate of the denominator, which is \( 1-\sqrt{2} \):

\[ \frac{1}{1+\sqrt{2}} \cdot \frac{1-\sqrt{2}}{1-\sqrt{2}} = \frac{1 - \sqrt{2}}{(1+\sqrt{2})(1-\sqrt{2})} \]

Calculating the denominator:

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

Thus, we have:

\[ \frac{1 - \sqrt{2}}{-1} = \sqrt{2} - 1 \]

### Step 2: Simplify \( \frac{1}{1-\sqrt{2}} \)
Similarly, we simplify \( \frac{1}{1-\sqrt{2}} \) by multiplying by the conjugate \( 1+\sqrt{2} \):

\[ \frac{1}{1-\sqrt{2}} \cdot \frac{1+\sqrt{2}}{1+\sqrt{2}} = \frac{1 + \sqrt{2}}{(1-\sqrt{2})(1+\sqrt{2})} \]

Calculating the denominator:

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

Thus, we have:

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

### Step 3: Substitute back into the original expression
Now we substitute back into the original expression:

\[ 1 - \left(\sqrt{2} - 1\right) + \left(-1 - \sqrt{2}\right) \]

This simplifies to:

\[ 1 - \sqrt{2} + 1 - 1 - \sqrt{2} \]

### Step 4: Combine like terms
Combining the terms gives us:

\[ 1 + 1 - 1 - \sqrt{2} - \sqrt{2} = 1 - 2\sqrt{2} \]

### Final Result
Thus, the simplified value of the expression \( 1 - \frac{1}{1+\sqrt{2}} + \frac{1}{1-\sqrt{2}} \) is:

\[ \boxed{1 - 2\sqrt{2}} \]