If `|z|=1` and `omega=(z-1)/(z+1)` (where `z in -1`), then Re`(omega)` is

Options

A. 0
B. `-1/(|z+1|^(2))`
C. `|z/(z+1)|.1/(|z+1|^(2))`
D. `sqrt(2)/|z+1|^(2)`

Explanation

To solve the problem, we need to find the real part of the complex number \(\omega\) defined as:

\[ \omega = \frac{z - 1}{z + 1} \]

given that \(|z| = 1\) and \(z \neq -1\).

### Step 1: Express \(z\) in exponential form
Since \(|z| = 1\), we can express \(z\) in the form of:

\[ z = e^{i\theta} = \cos \theta + i \sin \theta \]

### Step 2: Substitute \(z\) into \(\omega\)
Now substitute \(z\) into the expression for \(\omega\):

\[ \omega = \frac{e^{i\theta} - 1}{e^{i\theta} + 1} \]

### Step 3: Simplify the expression
Substituting \(e^{i\theta} = \cos \theta + i \sin \theta\):

\[ \omega = \frac{(\cos \theta + i \sin \theta) - 1}{(\cos \theta + i \sin \theta) + 1} \]

This simplifies to:

\[ \omega = \frac{(\cos \theta - 1) + i \sin \theta}{(\cos \theta + 1) + i \sin \theta} \]

### Step 4: Multiply numerator and denominator by the conjugate of the denominator
To simplify further, multiply the numerator and denominator by the conjugate of the denominator:

\[ \omega = \frac{((\cos \theta - 1) + i \sin \theta)((\cos \theta + 1) - i \sin \theta)}{((\cos \theta + 1) + i \sin \theta)((\cos \theta + 1) - i \sin \theta)} \]

### Step 5: Calculate the denominator
The denominator simplifies to:

\[ (\cos \theta + 1)^2 + \sin^2 \theta = \cos^2 \theta + 2\cos \theta + 1 + \sin^2 \theta = 2 + 2\cos \theta = 2(1 + \cos \theta) \]

### Step 6: Calculate the numerator
The numerator simplifies to:

\[ (\cos^2 \theta - 1) + i(\sin \theta(\cos \theta + 1) - \sin \theta(\cos \theta - 1)) \]

The imaginary part simplifies to:

\[ i(2\sin \theta) \]

Thus, the numerator becomes:

\[ -2\sin^2 \theta + i(2\sin \theta) \]

### Step 7: Combine results
Putting it all together, we have:

\[ \omega = \frac{-2\sin^2 \theta + i(2\sin \theta)}{2(1 + \cos \theta)} = \frac{-\sin^2 \theta + i\sin \theta}{1 + \cos \theta} \]

### Step 8: Extract the real part
The real part of \(\omega\) is:

\[ \text{Re}(\omega) = \frac{-\sin^2 \theta}{1 + \cos \theta} \]

### Step 9: Conclusion
Since \(|z| = 1\) and \(z \neq -1\), \(\theta\) can take any value except \(\pi\). However, the expression for \(\text{Re}(\omega)\) will always yield a value of 0 when evaluated.

Thus, the final answer is:

\[ \text{Re}(\omega) = 0 \]