What is the modulus of `|(1+2i)/(1-(1-i)^(2))|`?

Options

A. 1
B. `sqrt5`
C. `sqrt3`
D. 5

Explanation

To find the modulus of the expression \(\left|\frac{1 + 2i}{1 - (1 - i)^2}\right|\), we will follow these steps:

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

Calculating \((1 - i)^2\):
\[ (1 - i)^2 = 1^2 - 2 \cdot 1 \cdot i + i^2 = 1 - 2i + (-1) = 0 - 2i = -2i \]

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

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

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

### Step 4: Find the modulus
The modulus of a complex number \(z = a + bi\) is given by \(|z| = \sqrt{a^2 + b^2}\). In this case, since we have simplified the expression to \(1\), we can write:
\[ |1| = \sqrt{1^2 + 0^2} = \sqrt{1} = 1 \]

Thus, the modulus of the given expression is:
\[ \boxed{1} \]

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