If z is a point on the Argand plane such that `|z - 1| = 1 " thea " ( z - 2)/( z)` is equal to

Options

A. tan (arg z)
B. cot (arg z)
C. i tan (arg z)
D. none

Answer

To solve the problem, we start with the given condition and manipulate the expressions accordingly.

### Step-by-Step Solution:

1. **Understanding the Condition**:
We are given that \( |z - 1| = 1 \). This means that the point \( z \) lies on a circle in the Argand plane with center at \( 1 \) (which corresponds to the complex number \( 1 + 0i \)) and radius \( 1 \).

2. **Expressing \( z \)**:
We can express \( z \) in terms of a parameter \( \theta \) as follows:
\[ z = 1 + e^{i\theta} \]
where \( e^{i\theta} \) represents points on the unit circle.

3. **Finding \( z - 2 \)**:
Now, we calculate \( z - 2 \):
\[ z - 2 = (1 + e^{i\theta}) - 2 = e^{i\theta} - 1 \]

4. **Expressing \( z \) in a different form**:
We can also write \( z \) as:
\[ z = 1 + e^{i\theta} = 1 + (\cos\theta + i\sin\theta) = (1 + \cos\theta) + i\sin\theta \]

5. **Calculating \( \frac{z - 2}{z} \)**:
Now we need to find \( \frac{z - 2}{z} \):
\[ \frac{z - 2}{z} = \frac{e^{i\theta} - 1}{1 + e^{i\theta}} \]

6. **Simplifying the Expression**:
We can simplify this expression further:
\[ \frac{e^{i\theta} - 1}{1 + e^{i\theta}} = \frac{(\cos\theta - 1) + i\sin\theta}{(1 + \cos\theta) + i\sin\theta} \]

7. **Using Trigonometric Identities**:
We know that \( \cos\theta - 1 = -2\sin^2\left(\frac{\theta}{2}\right) \) and \( 1 + \cos\theta = 2\cos^2\left(\frac{\theta}{2}\right) \):
\[ \frac{-2\sin^2\left(\frac{\theta}{2}\right) + i\sin\theta}{2\cos^2\left(\frac{\theta}{2}\right) + i\sin\theta} \]

8. **Finding the Argument**:
The argument of the complex number can be found using the formula:
\[ \text{arg}(z) = \tan^{-1}\left(\frac{\text{Imaginary part}}{\text{Real part}}\right) \]

9. **Final Result**:
After performing the calculations, we find that:
\[ \frac{z - 2}{z} = -i \tan\left(\frac{\theta}{2}\right) \]
Hence, the argument of \( \frac{z - 2}{z} \) is \( -\frac{\theta}{2} \).

### Conclusion:
Thus, the final answer is that \( \frac{z - 2}{z} \) is equal to \( -i \tan\left(\frac{\theta}{2}\right) \).