If `a=sqrt(2)+1` and `b=sqrt(2)-1`, then the value of `(1)/(a+1)+(1)/(b+1)` will be

Options

A. 0
B. 1
C. 2
D. `-1`

Answer

To solve the problem, we need to find the value of \(\frac{1}{a+1} + \frac{1}{b+1}\) given that \(a = \sqrt{2} + 1\) and \(b = \sqrt{2} - 1\).

### Step-by-Step Solution:

1. **Substitute the values of \(a\) and \(b\)**:
\[ a + 1 = (\sqrt{2} + 1) + 1 = \sqrt{2} + 2 \]
\[ b + 1 = (\sqrt{2} - 1) + 1 = \sqrt{2} \]

2. **Rewrite the expression**:
We need to calculate:
\[ \frac{1}{a+1} + \frac{1}{b+1} = \frac{1}{\sqrt{2} + 2} + \frac{1}{\sqrt{2}} \]

3. **Find a common denominator**:
The common denominator for the two fractions is \((\sqrt{2} + 2) \cdot \sqrt{2}\). Thus, we rewrite the expression:
\[ \frac{\sqrt{2}}{(\sqrt{2} + 2) \cdot \sqrt{2}} + \frac{\sqrt{2} + 2}{(\sqrt{2} + 2) \cdot \sqrt{2}} \]

4. **Combine the fractions**:
\[ \frac{\sqrt{2} + (\sqrt{2} + 2)}{(\sqrt{2} + 2) \cdot \sqrt{2}} = \frac{2\sqrt{2} + 2}{(\sqrt{2} + 2) \cdot \sqrt{2}} \]

5. **Factor the numerator**:
\[ = \frac{2(\sqrt{2} + 1)}{(\sqrt{2} + 2) \cdot \sqrt{2}} \]

6. **Simplify the expression**:
Now we can simplify:
\[ = \frac{2(\sqrt{2} + 1)}{\sqrt{2}(\sqrt{2} + 2)} \]

7. **Evaluate the expression**:
We can substitute \(\sqrt{2} = 1.414\) to evaluate the expression numerically, but we can also notice that:
\[ = \frac{2(\sqrt{2} + 1)}{\sqrt{2}(\sqrt{2} + 2)} = 1 \]

Thus, the final value of \(\frac{1}{a+1} + \frac{1}{b+1}\) is:
\[ \boxed{1} \]