The argument of `(1-i)/(1+i)` is-

Options

A. `-pi/2 `
B. `pi/2 `
C. `(3pi)/2 `
D. `(5pi)/2`

Explanation

To find the argument of the complex number \((1-i)/(1+i)\), we can follow these steps:

### Step 1: Write the complex number
Let \( z = \frac{1-i}{1+i} \).

### Step 2: Multiply by the conjugate of the denominator
To simplify \( z \), we multiply the numerator and the denominator by the conjugate of the denominator, which is \( 1-i \):
\[ z = \frac{(1-i)(1-i)}{(1+i)(1-i)} \]

### Step 3: Expand the numerator and denominator
Now, we will expand both the numerator and the denominator:
- **Numerator**:
\[ (1-i)(1-i) = 1 - 2i + i^2 = 1 - 2i - 1 = -2i \]
- **Denominator**:
\[ (1+i)(1-i) = 1^2 - i^2 = 1 - (-1) = 1 + 1 = 2 \]

### Step 4: Simplify \( z \)
Now we can simplify \( z \):
\[ z = \frac{-2i}{2} = -i \]

### Step 5: Compare with standard form
The complex number \( -i \) can be expressed in the standard form \( a + bi \) where \( a = 0 \) and \( b = -1 \).

### Step 6: Determine the argument
To find the argument of \( z \), we need to locate it in the complex plane. Since \( a = 0 \) and \( b < 0 \), the point \( (0, -1) \) lies on the negative \( y \)-axis, which is in the fourth quadrant.

### Step 7: Calculate the argument
The argument \( \theta \) of a complex number in the fourth quadrant is given by:
\[ \theta = -\frac{\pi}{2} \]

### Final Answer
Thus, the argument of \( z \) is:
\[ \text{Argument of } z = -\frac{\pi}{2} \]