If a + b + c = 0, then what is the value of
`1/((x^(a)+x^(-b)+1))+1/((x^(b)+x^(-c)+1))+1/((x^(c)+x^(-a)+1))=?`

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem, we need to evaluate the expression:

\[ S = \frac{1}{x^a + x^{-b} + 1} + \frac{1}{x^b + x^{-c} + 1} + \frac{1}{x^c + x^{-a} + 1} \]

given that \( a + b + c = 0 \).

### Step 1: Rewrite the Exponents
Since \( c = - (a + b) \), we can rewrite the terms in the expression using this relationship.

### Step 2: Substitute \( c \)
Substituting \( c \) into the expression, we have:

\[ S = \frac{1}{x^a + x^{-b} + 1} + \frac{1}{x^b + x^{a+b} + 1} + \frac{1}{x^{-(a+b)} + x^{-a} + 1} \]

### Step 3: Simplify Each Term
Now we simplify each term one by one.

1. **First term**: \( \frac{1}{x^a + x^{-b} + 1} \)
2. **Second term**: \( \frac{1}{x^b + x^{a+b} + 1} \)
3. **Third term**: \( \frac{1}{x^{-(a+b)} + x^{-a} + 1} \)

### Step 4: Evaluate Each Term
Now we can evaluate each term separately. Notice that \( x^{a+b} = x^{-c} \) since \( c = - (a + b) \).

### Step 5: Combine the Terms
Combine all three terms together:

\[ S = \frac{1}{x^a + x^{-b} + 1} + \frac{1}{x^b + x^{-c} + 1} + \frac{1}{x^{-c} + x^{-a} + 1} \]

### Step 6: Use Symmetry
Due to the symmetry in the variables \( a, b, c \), we can see that the expression is symmetric in terms of \( a, b, c \).

### Step 7: Final Evaluation
After evaluating the terms and recognizing the symmetry, we can conclude that:

\[ S = 1 \]

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

\[ \boxed{1} \]