`[(1 + sin "" (pi)/( 8) + i cos "" (pi)/( 8))/( 1 + sin "" (pi)/( 8) - i cos "" (pi)/( 8))]^(8) = `

Options

A. `-1`
B. `1`
C. `i`
D. `-i`

Answer

To solve the given expression

\[ \left[\frac{1 + \sin\left(\frac{\pi}{8}\right) + i \cos\left(\frac{\pi}{8}\right)}{1 + \sin\left(\frac{\pi}{8}\right) - i \cos\left(\frac{\pi}{8}\right)}\right]^{8}, \]

we will follow these steps:

### Step 1: Simplify the Expression
We start with the expression inside the brackets:

\[ \frac{1 + \sin\left(\frac{\pi}{8}\right) + i \cos\left(\frac{\pi}{8}\right)}{1 + \sin\left(\frac{\pi}{8}\right) - i \cos\left(\frac{\pi}{8}\right)}. \]

Let \( a = 1 + \sin\left(\frac{\pi}{8}\right) \) and \( b = \cos\left(\frac{\pi}{8}\right) \). Thus, we can rewrite the expression as:

\[ \frac{a + ib}{a - ib}. \]

### Step 2: Multiply by the Conjugate
To simplify, we multiply the numerator and denominator by the conjugate of the denominator:

\[ \frac{(a + ib)(a + ib)}{(a - ib)(a + ib)} = \frac{(a + ib)^2}{a^2 + b^2}. \]

### Step 3: Calculate the Denominator
The denominator simplifies to:

\[ a^2 + b^2 = \left(1 + \sin\left(\frac{\pi}{8}\right)\right)^2 + \cos^2\left(\frac{\pi}{8}\right). \]

Expanding \( a^2 \):

\[ = 1 + 2\sin\left(\frac{\pi}{8}\right) + \sin^2\left(\frac{\pi}{8}\right) + \cos^2\left(\frac{\pi}{8}\right). \]

Using the identity \( \sin^2 x + \cos^2 x = 1 \):

\[ = 1 + 2\sin\left(\frac{\pi}{8}\right) + 1 = 2 + 2\sin\left(\frac{\pi}{8}\right). \]

### Step 4: Calculate the Numerator
Now for the numerator:

\[ (a + ib)^2 = (1 + \sin\left(\frac{\pi}{8}\right) + i \cos\left(\frac{\pi}{8}\right))^2. \]

Expanding this gives:

\[ = (1 + \sin\left(\frac{\pi}{8}\right))^2 + 2i(1 + \sin\left(\frac{\pi}{8}\right))\cos\left(\frac{\pi}{8}\right) - \cos^2\left(\frac{\pi}{8}\right). \]

### Step 5: Combine and Simplify
Now we have:

\[ \frac{(1 + \sin\left(\frac{\pi}{8}\right))^2 - \cos^2\left(\frac{\pi}{8}\right) + 2i(1 + \sin\left(\frac{\pi}{8}\right))\cos\left(\frac{\pi}{8}\right)}{2 + 2\sin\left(\frac{\pi}{8}\right)}. \]

### Step 6: Convert to Exponential Form
Using Euler's formula, we can express this in the form \( e^{i\theta} \). The angle \( \theta \) can be determined from the argument of the complex number.

### Step 7: Raise to the Power of 8
After simplifying to the form \( e^{i\theta} \), we raise it to the power of 8:

\[ \left(e^{i\theta}\right)^{8} = e^{8i\theta}. \]

### Step 8: Find the Final Value
The final value will be \( e^{8i\theta} \), which corresponds to \( \cos(8\theta) + i\sin(8\theta) \).

### Step 9: Evaluate \( \cos(3\pi) \) and \( \sin(3\pi) \)
From the calculations, we find that \( 8\theta = 3\pi \):

- \( \cos(3\pi) = -1 \)
- \( \sin(3\pi) = 0 \)

Thus, the final answer is:

\[ \boxed{-1}. \]