The value of `(sin frac(pi)(8) + i cos frac(pi)(8))^(8)/((sin frac(pi)(8) - i cos frac(pi)(8))^(8))` is :

Options

A. `-1`
B. 0
C. 1
D. `2i`

Answer

To solve the problem, we need to evaluate the expression:

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

### Step 1: Rewrite the expression
We can rewrite the numerator and denominator using the property of complex numbers:

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

Thus, we can express the numerator as:

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

### Step 2: Factor out \(i^8\)
Now, we can factor out \(i^8\) from the expression:

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

Since \(i^8 = 1\), we have:

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

### Step 3: Rewrite the denominator
Similarly, we can rewrite the denominator:

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

Factoring out \((-i)^8\):

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

Since \((-i)^8 = 1\), we have:

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

### Step 4: Combine the results
Now, substituting back into our original expression, we get:

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

### Conclusion
Thus, the value of the given expression is:

\[ \boxed{1} \]