The modulus and amplitude of `(1+2i)/(1-(1-i)^(2))` are

Options

A. `sqrt(2)` and `(pi)/(6)`
B. `1` and `(pi)/(4)`
C. `1` and `0`
D. `1` and `(pi)/(3)`

Explanation

To find the modulus and amplitude of the complex number \(\frac{1+2i}{1-(1-i)^2}\), we will follow these steps:

### Step 1: Simplify the denominator
First, we need to simplify the expression in the denominator, \(1 - (1 - i)^2\).

Calculating \((1 - i)^2\):
\[ (1 - i)^2 = 1^2 - 2 \cdot 1 \cdot i + i^2 = 1 - 2i + (-1) = -2i \]
Now substituting this back into the denominator:
\[ 1 - (1 - i)^2 = 1 - (-2i) = 1 + 2i \]

### Step 2: Rewrite the complex number
Now we can rewrite the complex number:
\[ \frac{1 + 2i}{1 + 2i} \]

### Step 3: Simplify the fraction
Since the numerator and denominator are the same, we can simplify:
\[ \frac{1 + 2i}{1 + 2i} = 1 \]

### Step 4: Find the modulus
The modulus of a real number \(1\) is simply:
\[ |1| = 1 \]

### Step 5: Find the amplitude
The amplitude (or argument) of a real number is:
\[ \text{arg}(1) = 0 \]

### Final Result
Thus, the modulus and amplitude of the given complex number are:
- Modulus: \(1\)
- Amplitude: \(0\)