$(\frac{1+cos\frac{\pi}{8}+i~sin\frac{\pi}{8}}{1+cos\frac{\pi}{8}-i~sin\frac{\pi}{8}})^{8}=$

Question

Accepted Answer

To solve the given expression $(\frac{1+\cos\frac{\pi}{8}+i~\sin\frac{\pi}{8}}{1+\cos\frac{\pi}{8}-i~\sin\frac{\pi}{8}})^{8}$, we need to simplify the fraction inside the parentheses first.

### Step-by-Step Explanation:

1. **Understanding the Expression:**
   The expression inside the parentheses is a complex fraction. It can be simplified using the properties of complex numbers and Euler's formula.

2. **Simplifying the Fraction:**
   The numerator is $1 + \cos\frac{\pi}{8} + i \sin\frac{\pi}{8}$ and the denominator is $1 + \cos\frac{\pi}{8} - i \sin\frac{\pi}{8}$.

3. **Using Euler's Formula:**
   Recall that Euler's formula states $e^{i\theta} = \cos\theta + i\sin\theta$. Therefore, the numerator can be seen as a complex number in polar form: $1 + \cos\frac{\pi}{8} + i \sin\frac{\pi}{8} = e^{i\theta_1}$ and the denominator as $1 + \cos\frac{\pi}{8} - i \sin\frac{\pi}{8} = e^{-i\theta_1}$.

4. **Complex Conjugate:**
   The expression $\frac{1+\cos\frac{\pi}{8}+i~\sin\frac{\pi}{8}}{1+\cos\frac{\pi}{8}-i~\sin\frac{\pi}{8}}$ can be simplified by multiplying the numerator and the denominator by the complex conjugate of the denominator:
   $$
   \frac{(1+\cos\frac{\pi}{8}+i~\sin\frac{\pi}{8})(1+\cos\frac{\pi}{8}+i~\sin\frac{\pi}{8})}{(1+\cos\frac{\pi}{8})^2 + (\sin\frac{\pi}{8})^2}
   $$

5. **Simplifying Further:**
   The denominator simplifies to $1 + 2\cos\frac{\pi}{8} + \cos^2\frac{\pi}{8} + \sin^2\frac{\pi}{8} = 2(1 + \cos\frac{\pi}{8})$.

6. **Resulting Expression:**
   The fraction simplifies to a complex number of the form $e^{i\theta}$, where $\theta$ is related to the angle $\frac{\pi}{8}$.

7. **Raising to the Power of 8:**
   When you raise this expression to the power of 8, you effectively multiply the angle by 8:
   $$
   \left(e^{i\theta}\right)^8 = e^{i(8\theta)}
   $$

8. **Evaluating the Angle:**
   Since the angle $\theta$ is such that $8\theta = \pi$, we have:
   $$
   e^{i\pi} = -1
   $$

### Conclusion:

- **Correct Answer: Option 2 (-1):** The expression simplifies to $-1$ after raising to the power of 8, which matches Option 2.
- **Why Other Options are Incorrect:**
  - **Option 1 (1):** This would imply the angle is a multiple of $2\pi$, which is not the case here.
  - **Option 3 (2) and Option 4 (-2):** These are not possible results for the given expression as it simplifies to a unit circle value, specifically $-1$.

Thus, the correct answer is indeed Option 2, $-1$.