The value of `(1)/(1+sqrt(2)+sqrt(3))+(1)/(1-sqrt(2)+sqrt(3))` is :

Options

A. A) `sqrt(2)`
B. B) `sqrt(3)`
C. C) `1`
D. D) `4(sqrt(3)+sqrt(2))`

Answer

To solve the expression

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

we will follow these steps:

### Step 1: Identify the common denominator
The common denominator for the two fractions is the product of the denominators:

\[ (1 + \sqrt{2} + \sqrt{3})(1 - \sqrt{2} + \sqrt{3}). \]

### Step 2: Rewrite the fractions
We can rewrite the fractions with the common denominator:

\[ \frac{(1 - \sqrt{2} + \sqrt{3}) + (1 + \sqrt{2} + \sqrt{3})}{(1 + \sqrt{2} + \sqrt{3})(1 - \sqrt{2} + \sqrt{3})}. \]

### Step 3: Simplify the numerator
Now, simplify the numerator:

\[ (1 - \sqrt{2} + \sqrt{3}) + (1 + \sqrt{2} + \sqrt{3}) = 2 + 2\sqrt{3}. \]

### Step 4: Simplify the denominator
Next, we simplify the denominator using the difference of squares:

\[ (1 + \sqrt{2} + \sqrt{3})(1 - \sqrt{2} + \sqrt{3}) = (1 + \sqrt{3})^2 - (\sqrt{2})^2. \]

Calculating this gives:

\[ (1 + \sqrt{3})^2 = 1 + 2\sqrt{3} + 3 = 4 + 2\sqrt{3}, \]
and
\[ (\sqrt{2})^2 = 2. \]

Thus, the denominator becomes:

\[ (4 + 2\sqrt{3}) - 2 = 2 + 2\sqrt{3}. \]

### Step 5: Combine the results
Now we can combine the results:

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

### Final Answer
Thus, the value of the expression is:

\[ \boxed{1}. \]