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 one 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. **Understanding the Condition**: Since \( |z| = 1 \), we can express \( z \) in its polar form as \( z = e^{i\theta} \) where \( \theta = \arg(z) \).

2. **Expressing \( z+1 \) and \( z-1 \)**:
- \( z + 1 = e^{i\theta} + 1 \)
- \( z - 1 = e^{i\theta} - 1 \)

3. **Using the Argument Property**:
We can use the property of arguments:
\[ \arg(a) + \arg(b) = \arg(ab) \]
Therefore,
\[ \arg(z) - \arg(z+1) - \arg(z-1) = \arg\left(\frac{z}{(z+1)(z-1)}\right) \]

4. **Calculating the Product**:
\[ (z+1)(z-1) = z^2 - 1 \]
Thus, we can rewrite the expression as:
\[ \arg\left(\frac{z}{z^2 - 1}\right) \]

5. **Finding the Modulus**:
Since \( |z| = 1 \), we have \( |z^2| = |z|^2 = 1 \). Therefore, \( |z^2 - 1| = |z^2 - 1| = |(z^2 - 1)| \).

6. **Simplifying the Argument**:
We can express the argument as:
\[ \arg(z) - \arg(z^2 - 1) = \arg(z) - \arg(z^2 - 1) \]
Since \( z^2 - 1 = (z - 1)(z + 1) \), we can further simplify:
\[ \arg(z^2 - 1) = \arg(z - 1) + \arg(z + 1) \]

7. **Final Expression**:
Therefore, we have:
\[ \arg(z) - \arg(z + 1) - \arg(z - 1) = \arg(z) - (\arg(z - 1) + \arg(z + 1)) \]

8. **Finding a Possible Value**:
Since \( z \) lies on the unit circle and \( z \neq \pm 1 \), we can choose a specific value for \( z \) to find a possible value for the expression. For example, let \( z = i \) (which corresponds to \( \theta = \frac{\pi}{2} \)):
- \( \arg(i) = \frac{\pi}{2} \)
- \( z + 1 = i + 1 \) and \( z - 1 = i - 1 \)

We can calculate:
- \( \arg(z + 1) = \arg(i + 1) \)
- \( \arg(z - 1) = \arg(i - 1) \)

After calculating these arguments, we can find the value of \( \arg(z) - \arg(z + 1) - \arg(z - 1) \).

### Conclusion:
After performing the calculations, we find that one possible value of \( \arg(z) - \arg(z + 1) - \arg(z - 1) \) is \( -\frac{\pi}{2} \).