If z is a complex number of unit modulus and argument q, then `a r g((1+z)/(1+ bar z))` equal (1) `pi/2-theta` (2) `theta` (3) `pi-theta` (4) `-theta`

Options

A. `-theta`
B. `(pi)/(2 ) - theta`
C. `theta`
D. `pi-theta`

Explanation

To solve the problem, we need to find the argument of the expression \(\frac{1+z}{1+\bar{z}}\) where \(z\) is a complex number of unit modulus and argument \(\theta\).

### Step-by-step Solution:

1. **Understanding the properties of \(z\)**:
Since \(z\) is a complex number of unit modulus, we can express it in the polar form:
\[ z = e^{i\theta} \]
where \(\theta\) is the argument of \(z\).

2. **Finding the conjugate of \(z\)**:
The conjugate of \(z\) is given by:
\[ \bar{z} = e^{-i\theta} \]

3. **Substituting \(z\) and \(\bar{z}\) into the expression**:
We need to compute:
\[ \frac{1+z}{1+\bar{z}} = \frac{1 + e^{i\theta}}{1 + e^{-i\theta}} \]

4. **Simplifying the expression**:
We can multiply the numerator and the denominator by \(e^{i\theta}\) to eliminate the complex exponentials:
\[ = \frac{e^{i\theta} + 1}{e^{i\theta} + 1} = \frac{1 + e^{i\theta}}{1 + e^{-i\theta}} = \frac{(1 + e^{i\theta})(e^{i\theta})}{(1 + e^{-i\theta})(e^{i\theta})} \]
This simplifies to:
\[ = \frac{e^{i\theta} + 1}{e^{i\theta} + 1} \]

5. **Finding the argument**:
Since both the numerator and denominator are equal, we can conclude:
\[ \arg\left(\frac{1+z}{1+\bar{z}}\right) = \arg(1) = 0 \]

6. **Final answer**:
Therefore, the argument of \(\frac{1+z}{1+\bar{z}}\) is:
\[ \arg\left(\frac{1+z}{1+\bar{z}}\right) = \theta \]

### Conclusion:
The correct option is (2) \(\theta\).

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