On simplification the value of `1- (1)/( 1 + sqrt2) + (1)/( 1 - sqrt2)` is

Options

A. `2 sqrt2 -1`
B. `1 - 2 sqrt2`
C. `1 - sqrt2`
D. `-2 sqrt2`

Answer

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

### Step 1: Write down the expression
We start with the expression:
\[ 1 - \frac{1}{1 + \sqrt{2}} + \frac{1}{1 - \sqrt{2}} \]

### Step 2: Find a common denominator
To combine the fractions, we need a common denominator. The common denominator for \(1 + \sqrt{2}\) and \(1 - \sqrt{2}\) is \((1 + \sqrt{2})(1 - \sqrt{2})\), which simplifies to \(1 - 2 = -1\).

### Step 3: Rewrite the fractions
Now we can rewrite the fractions with the common denominator:
\[ 1 - \frac{(1 - \sqrt{2})}{(1 + \sqrt{2})(1 - \sqrt{2})} + \frac{(1 + \sqrt{2})}{(1 + \sqrt{2})(1 - \sqrt{2})} \]
This simplifies to:
\[ 1 - \frac{(1 - \sqrt{2}) + (1 + \sqrt{2})}{-1} \]

### Step 4: Simplify the numerator
Now, we simplify the numerator:
\[ (1 - \sqrt{2}) + (1 + \sqrt{2}) = 1 - \sqrt{2} + 1 + \sqrt{2} = 2 \]
Thus, we have:
\[ 1 - \frac{2}{-1} = 1 + 2 = 3 \]

### Step 5: Final simplification
Now, we go back to the expression:
\[ 1 - \frac{2}{-1} = 1 + 2 = 3 \]

### Conclusion
The final value of the expression is:
\[ \boxed{3} \]