If `|z|=1` and `z ne +-1`, then one of the possible value of `arg(z)-arg(z+1)-arg(z-1)` , is

Options

A. `-pi//6`
B. `pi//3`
C. `-pi//2`
D. `pi//4`

Explanation

To solve the problem, we need to find a possible value of the expression \( \arg(z) - \arg(z+1) - \arg(z-1) \) given that \( |z| = 1 \) and \( z \neq \pm 1 \).

### Step-by-Step Solution:

1. **Use the property of arguments**:
We can use the property of arguments that states:
\[ \arg(z_1) - \arg(z_2) = \arg\left(\frac{z_1}{z_2}\right) \]
Therefore, we can rewrite the expression as:
\[ \arg(z) - \arg(z+1) - \arg(z-1) = \arg\left(\frac{z}{(z+1)(z-1)}\right) \]

2. **Simplify the expression**:
The expression inside the argument can be simplified:
\[ (z+1)(z-1) = z^2 - 1 \]
Thus, we have:
\[ \arg\left(\frac{z}{z^2 - 1}\right) \]

3. **Use the modulus condition**:
Since \( |z| = 1 \), we know that \( z \bar{z} = 1 \). Therefore, \( z^2 \bar{z}^2 = 1 \) implies \( |z^2| = 1 \) as well. This means:
\[ |z^2 - 1| = |z^2| \cdot |1| = 1 \]
We can express \( z^2 - 1 \) in terms of \( z \) and its conjugate:
\[ |z^2 - 1| = |z - 1||z + 1| \]

4. **Express \( z^2 - 1 \)**:
We can write:
\[ z^2 - 1 = z^2 - z \bar{z} = z(z - \bar{z}) \]
where \( z - \bar{z} = 2i \text{Im}(z) \) (as shown in the video).

5. **Find the argument**:
Since \( z - \bar{z} = 2i \text{Im}(z) \), we can express:
\[ \arg(z - \bar{z}) = \arg(2i \text{Im}(z)) = \frac{\pi}{2} \text{ or } -\frac{\pi}{2} \]
Thus, we have:
\[ \arg\left(\frac{z}{z^2 - 1}\right) = \arg(z) - \arg(z^2 - 1) \]

6. **Final value**:
Since \( z \) lies on the unit circle, \( \arg(z) \) can be any angle \( \theta \). Therefore:
\[ \arg(z) - \arg(z^2 - 1) = \theta - \left(\frac{\pi}{2} \text{ or } -\frac{\pi}{2}\right) \]
This means the possible values of \( \arg(z) - \arg(z+1) - \arg(z-1) \) can be \( \frac{\pi}{2} \) or \( -\frac{\pi}{2} \).

7. **Conclusion**:
Since the question asks for one of the possible values, we can conclude that:
\[ \text{One of the possible values is } -\frac{\pi}{2}. \]