What is value of `[(1)/(1-x^((p-q)))+(1)/(1-x^((q-p)))]`?

Options

A. 0
B. 1
C. `(x^(q)-x^(p)//(x^(q)+x^(p))`
D. `(x^(q)+x^(p))//(x^(q)-x^(p))`

Answer

To solve the expression \(\left[\frac{1}{1 - x^{(p - q)}} + \frac{1}{1 - x^{(q - p)}}\right]\), we will follow these steps:

### Step 1: Rewrite the Expression
We start with the given expression:
\[ \frac{1}{1 - x^{(p - q)}} + \frac{1}{1 - x^{(q - p)}} \]

### Step 2: Identify the Terms
Notice that \(x^{(q - p)}\) can be rewritten as \(\frac{1}{x^{(p - q)}}\). Thus, we can rewrite the second term:
\[ \frac{1}{1 - x^{(q - p)}} = \frac{1}{1 - \frac{1}{x^{(p - q)}}} \]

### Step 3: Simplify the Second Term
To simplify \(\frac{1}{1 - \frac{1}{x^{(p - q)}}}\), we can find a common denominator:
\[ 1 - \frac{1}{x^{(p - q)}} = \frac{x^{(p - q)} - 1}{x^{(p - q)}} \]
Thus, the second term becomes:
\[ \frac{x^{(p - q)}}{x^{(p - q)} - 1} \]

### Step 4: Combine the Two Terms
Now we have:
\[ \frac{1}{1 - x^{(p - q)}} + \frac{x^{(p - q)}}{x^{(p - q)} - 1} \]
We can find a common denominator for these two fractions:
\[ (1 - x^{(p - q)})(x^{(p - q)} - 1) \]

### Step 5: Write the Combined Fraction
The combined fraction becomes:
\[ \frac{(x^{(p - q)} - 1) + (1 - x^{(p - q)})}{(1 - x^{(p - q)})(x^{(p - q)} - 1)} \]
The numerator simplifies to:
\[ x^{(p - q)} - 1 + 1 - x^{(p - q)} = 0 \]

### Step 6: Conclusion
Since the numerator is zero, the entire expression evaluates to zero:
\[ \frac{0}{(1 - x^{(p - q)})(x^{(p - q)} - 1)} = 0 \]

Thus, the value of the expression is:
\[ \boxed{0} \]