The value of `| (1 + i sqrt(3))/(( 1 + (1)/( i + 1))^(2))|` is

Options

A. `(5)/(4)`
B. `(4)/(5)`
C. 20
D. `(9)/(4)`

Answer

To solve the problem, we need to find the value of

\[ \left| \frac{1 + i \sqrt{3}}{\left( 1 + \frac{1}{i + 1} \right)^2} \right|. \]

### Step 1: Simplify the denominator

First, we simplify the expression in the denominator:

\[ 1 + \frac{1}{i + 1}. \]

To simplify \(\frac{1}{i + 1}\), we multiply the numerator and the denominator by the conjugate of the denominator:

\[ \frac{1}{i + 1} \cdot \frac{i - 1}{i - 1} = \frac{i - 1}{(i + 1)(i - 1)} = \frac{i - 1}{i^2 - 1} = \frac{i - 1}{-1 - 1} = \frac{i - 1}{-2} = -\frac{1}{2} + \frac{i}{2}. \]

Now substituting this back, we have:

\[ 1 + \left(-\frac{1}{2} + \frac{i}{2}\right) = \frac{1}{2} + \frac{i}{2}. \]

### Step 2: Square the result

Now we square this result:

\[ \left(\frac{1}{2} + \frac{i}{2}\right)^2 = \left(\frac{1}{2}\right)^2 + 2 \cdot \frac{1}{2} \cdot \frac{i}{2} + \left(\frac{i}{2}\right)^2 = \frac{1}{4} + \frac{i}{2} - \frac{1}{4} = \frac{i}{2}. \]

### Step 3: Substitute back into the original expression

Now we substitute this back into the original expression:

\[ \left| \frac{1 + i \sqrt{3}}{\frac{i}{2}} \right| = \left| \frac{2(1 + i \sqrt{3})}{i} \right|. \]

### Step 4: Simplify the fraction

Now we can simplify:

\[ \frac{2(1 + i \sqrt{3})}{i} = 2 \left( \frac{1}{i} + \sqrt{3} \right) = 2 \left( -i + \sqrt{3} \right) = 2\sqrt{3} - 2i. \]

### Step 5: Find the modulus

Now we find the modulus of \(2\sqrt{3} - 2i\):

\[ |2\sqrt{3} - 2i| = \sqrt{(2\sqrt{3})^2 + (-2)^2} = \sqrt{12 + 4} = \sqrt{16} = 4. \]

### Step 6: Final calculation

Now we divide this by the modulus of the denominator, which is:

\[ \left| \frac{i}{2} \right| = \frac{1}{2}. \]

Thus, the final value is:

\[ \frac{4}{\frac{1}{2}} = 4 \times 2 = 8. \]

### Final Answer

The value of

\[ \left| \frac{1 + i \sqrt{3}}{(1 + \frac{1}{i + 1})^2} \right| = 8. \]