If `x= (1)/(sqrt3 + sqrt2), y= (1)/(sqrt3- sqrt2)` then find `(1)/(x +1) + (1)/(y+ 1)`

Options

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

Answer

To solve the problem, we need to find the value of \( \frac{1}{x + 1} + \frac{1}{y + 1} \) given \( x = \frac{1}{\sqrt{3} + \sqrt{2}} \) and \( y = \frac{1}{\sqrt{3} - \sqrt{2}} \).

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

Given:
\[ x = \frac{1}{\sqrt{3} + \sqrt{2}} \]
To rationalize \( x \), we multiply the numerator and the denominator by \( \sqrt{3} - \sqrt{2} \):
\[ x = \frac{1 \cdot (\sqrt{3} - \sqrt{2})}{(\sqrt{3} + \sqrt{2})(\sqrt{3} - \sqrt{2})} \]
Using the difference of squares:
\[ (\sqrt{3})^2 - (\sqrt{2})^2 = 3 - 2 = 1 \]
Thus:
\[ x = \sqrt{3} - \sqrt{2} \]

### Step 2: Rationalize \( y \)

Given:
\[ y = \frac{1}{\sqrt{3} - \sqrt{2}} \]
To rationalize \( y \), we multiply the numerator and the denominator by \( \sqrt{3} + \sqrt{2} \):
\[ y = \frac{1 \cdot (\sqrt{3} + \sqrt{2})}{(\sqrt{3} - \sqrt{2})(\sqrt{3} + \sqrt{2})} \]
Again using the difference of squares:
\[ (\sqrt{3})^2 - (\sqrt{2})^2 = 3 - 2 = 1 \]
Thus:
\[ y = \sqrt{3} + \sqrt{2} \]

### Step 3: Calculate \( x + 1 \) and \( y + 1 \)

Now we calculate:
\[ x + 1 = (\sqrt{3} - \sqrt{2}) + 1 = \sqrt{3} - \sqrt{2} + 1 \]
\[ y + 1 = (\sqrt{3} + \sqrt{2}) + 1 = \sqrt{3} + \sqrt{2} + 1 \]

### Step 4: Find \( \frac{1}{x + 1} + \frac{1}{y + 1} \)

We need to find:
\[ \frac{1}{x + 1} + \frac{1}{y + 1} = \frac{1}{\sqrt{3} - \sqrt{2} + 1} + \frac{1}{\sqrt{3} + \sqrt{2} + 1} \]
To combine these fractions, we find a common denominator:
\[ \text{Common Denominator} = (\sqrt{3} - \sqrt{2} + 1)(\sqrt{3} + \sqrt{2} + 1) \]
The numerator will be:
\[ (\sqrt{3} + \sqrt{2} + 1) + (\sqrt{3} - \sqrt{2} + 1) \]
Simplifying the numerator:
\[ = 2\sqrt{3} + 2 \]

### Step 5: Simplify the expression

Now we have:
\[ \frac{2\sqrt{3} + 2}{(\sqrt{3} - \sqrt{2} + 1)(\sqrt{3} + \sqrt{2} + 1)} \]
Now, we need to calculate the denominator:
\[ (\sqrt{3} - \sqrt{2} + 1)(\sqrt{3} + \sqrt{2} + 1) = (\sqrt{3})^2 + \sqrt{3} + \sqrt{2} - \sqrt{2} + \sqrt{3} + 1 - \sqrt{2} + 1 \]
This simplifies to:
\[ = 3 + 2\sqrt{3} + 1 - 2 = 2 + 2\sqrt{3} \]

### Final Result

Thus, we have:
\[ \frac{2(\sqrt{3} + 1)}{2 + 2\sqrt{3}} = \frac{\sqrt{3} + 1}{1 + \sqrt{3}} = 1 \]
So:
\[ \frac{1}{x + 1} + \frac{1}{y + 1} = 1 \]