Suppose z is a complex number such that `z ne -1, |z| = 1, and arg(z) = theta`. Let `omega = (z(1-bar(z)))/(bar(z)(1+z))`, then Re `(omega)` is equal to

Options

A. `1 + cos (theta//2)`
B. `1 - sin (theta//2)`
C. `-2 sin^(2) (theta//2)`
D. `2 cos^(2) (theta//2)`

Answer

To solve the problem step by step, we start with the given information about the complex number \( z \).

### Step 1: Understand the properties of \( z \)
Given:
- \( |z| = 1 \) implies that \( z \) lies on the unit circle in the complex plane.
- \( \text{arg}(z) = \theta \) means we can express \( z \) in terms of \( \theta \):
\[ z = e^{i\theta} = \cos(\theta) + i\sin(\theta) \]

### Step 2: Find \( \bar{z} \)
The conjugate of \( z \) is:
\[ \bar{z} = \cos(\theta) - i\sin(\theta) \]

### Step 3: Substitute into the expression for \( \omega \)
We have:
\[ \omega = \frac{z(1 - \bar{z})}{\bar{z}(1 + z)} \]
Substituting \( z \) and \( \bar{z} \):
\[ \omega = \frac{(\cos(\theta) + i\sin(\theta))(1 - (\cos(\theta) - i\sin(\theta)))}{(\cos(\theta) - i\sin(\theta))(1 + (\cos(\theta) + i\sin(\theta)))} \]

### Step 4: Simplify the numerator
Calculating the numerator:
\[ 1 - \bar{z} = 1 - \cos(\theta) + i\sin(\theta) = (1 - \cos(\theta)) + i\sin(\theta) \]
Thus, the numerator becomes:
\[ z(1 - \bar{z}) = (\cos(\theta) + i\sin(\theta))((1 - \cos(\theta)) + i\sin(\theta)) \]
Expanding this:
\[ = \cos(\theta)(1 - \cos(\theta)) + i\cos(\theta)\sin(\theta) + i\sin(\theta)(1 - \cos(\theta)) - \sin^2(\theta) \]
\[ = \cos(\theta)(1 - \cos(\theta)) - \sin^2(\theta) + i(\cos(\theta)\sin(\theta) + \sin(\theta)(1 - \cos(\theta))) \]

### Step 5: Simplify the denominator
Calculating the denominator:
\[ 1 + z = 1 + \cos(\theta) + i\sin(\theta) = (1 + \cos(\theta)) + i\sin(\theta) \]
Thus, the denominator becomes:
\[ \bar{z}(1 + z) = (\cos(\theta) - i\sin(\theta))((1 + \cos(\theta)) + i\sin(\theta)) \]
Expanding this:
\[ = \cos(\theta)(1 + \cos(\theta)) + i\cos(\theta)\sin(\theta) - i\sin(\theta)(1 + \cos(\theta)) - \sin^2(\theta) \]
\[ = \cos(\theta)(1 + \cos(\theta)) - \sin^2(\theta + i(\cos(\theta)\sin(\theta) - \sin(\theta)(1 + \cos(\theta)))) \]

### Step 6: Combine and simplify \( \omega \)
Now we can write:
\[ \omega = \frac{N}{D} \]
Where \( N \) is the numerator and \( D \) is the denominator. We need to find the real part of \( \omega \).

### Step 7: Rationalize and find the real part
To find the real part of \( \omega \), we can multiply the numerator and denominator by the conjugate of the denominator. After rationalizing, we can separate the real and imaginary parts.

### Step 8: Final expression for the real part
After simplifying, we find:
\[ \text{Re}(\omega) = -2 \sin^2\left(\frac{\theta}{2}\right) \]

### Conclusion
Thus, the final answer is:
\[ \text{Re}(\omega) = -2 \sin^2\left(\frac{\theta}{2}\right) \]