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

Options

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

Answer

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 \(\omega\)
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:
- \(1 + \omega + \omega^2 = 0\)
- \(\omega^3 = 1\)

### Step 2: Rewrite the expression
We start with the expression:
\[ 1 - \frac{1}{1 + \omega} - \frac{1}{1 + \omega^2} \]

### Step 3: Find a common denominator
The common denominator for the fractions is \((1 + \omega)(1 + \omega^2)\). Thus, we rewrite the expression:
\[ 1 - \left( \frac{(1 + \omega^2) + (1 + \omega)}{(1 + \omega)(1 + \omega^2)} \right) \]

### Step 4: Simplify the numerator
Now, simplify the numerator:
\[ (1 + \omega^2) + (1 + \omega) = 2 + \omega + \omega^2 \]
Using the property \(1 + \omega + \omega^2 = 0\), we find:
\[ \omega + \omega^2 = -1 \]
Thus, the numerator becomes:
\[ 2 - 1 = 1 \]

### Step 5: Substitute back into the expression
Now substitute back into the expression:
\[ 1 - \frac{1}{(1 + \omega)(1 + \omega^2)} \]

### Step 6: Calculate \(1 + \omega\) and \(1 + \omega^2\)
Using the properties:
\[ 1 + \omega = -\omega^2 \quad \text{and} \quad 1 + \omega^2 = -\omega \]
Thus:
\[ (1 + \omega)(1 + \omega^2) = (-\omega^2)(-\omega) = \omega^3 = 1 \]

### Step 7: Final calculation
Now substitute this back:
\[ 1 - 1 = 0 \]

### Conclusion
The value of the expression \( 1 - \frac{1}{1 + \omega} - \frac{1}{1 + \omega^2} \) is:
\[ \boxed{0} \]