If `omega` is a complex cube root of unity, then what is the value of `1-(1)/((1+omega))-(1)/((1+omega^(2)))` ?

Options

A. 1
B. 0
C. `omega`
D. `omega^(2)`

Explanation

To solve the expression \( 1 - \frac{1}{1 + \omega} - \frac{1}{1 + \omega^2} \), where \( \omega \) is a complex cube root of unity, we can follow these steps:

### Step 1: Understand the properties of cube roots of unity
The complex cube roots of unity are \( 1, \omega, \) and \( \omega^2 \), where:
- \( \omega = e^{2\pi i / 3} \)
- \( \omega^2 = e^{4\pi i / 3} \)
- The properties include \( 1 + \omega + \omega^2 = 0 \) and \( \omega^3 = 1 \).

### Step 2: Substitute the values of \( 1 + \omega \) and \( 1 + \omega^2 \)
Using the property \( 1 + \omega + \omega^2 = 0 \), we can express:
- \( 1 + \omega = -\omega^2 \)
- \( 1 + \omega^2 = -\omega \)

### Step 3: Rewrite the expression
Now we can rewrite the expression:
\[ 1 - \frac{1}{1 + \omega} - \frac{1}{1 + \omega^2} = 1 - \frac{1}{-\omega^2} - \frac{1}{-\omega} \]
This simplifies to:
\[ 1 + \frac{1}{\omega^2} + \frac{1}{\omega} \]

### Step 4: Find a common denominator
The common denominator for the fractions \( \frac{1}{\omega^2} \) and \( \frac{1}{\omega} \) is \( \omega^2 \):
\[ 1 + \frac{1}{\omega^2} + \frac{1}{\omega} = 1 + \frac{1 + \omega}{\omega^2} \]

### Step 5: Substitute \( 1 + \omega \)
Using the property \( 1 + \omega + \omega^2 = 0 \), we have \( 1 + \omega = -\omega^2 \):
\[ 1 + \frac{-\omega^2}{\omega^2} = 1 - 1 = 0 \]

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