Find the value of following expression: `[(1-cospi/(10)+isinpi/(10))/(1-cospi/(10)-isinpi/(10))]^(10)`

Explanation

To solve the expression

\[ \left[\frac{1 - \cos\left(\frac{\pi}{10}\right) + i \sin\left(\frac{\pi}{10}\right)}{1 - \cos\left(\frac{\pi}{10}\right) - i \sin\left(\frac{\pi}{10}\right)}\right]^{10}, \]

we will follow these steps:

### Step 1: Define \( z \)
Let \( z = \cos\left(\frac{\pi}{10}\right) + i \sin\left(\frac{\pi}{10}\right) \). This can also be expressed using Euler's formula as \( z = e^{i \frac{\pi}{10}} \).

### Step 2: Find \( \frac{1 - z}{1 - \frac{1}{z}} \)
We can rewrite the expression as:

\[ \frac{1 - z}{1 - \frac{1}{z}} = \frac{1 - z}{\frac{z - 1}{z}} = \frac{(1 - z)z}{z - 1}. \]

### Step 3: Simplify the expression
We can simplify this further:

\[ \frac{(1 - z)z}{z - 1} = -\frac{(z - 1)z}{z - 1} = -z. \]

### Step 4: Raise to the power of 10
Now, we raise the result to the power of 10:

\[ (-z)^{10} = (-1)^{10} z^{10} = z^{10}. \]

### Step 5: Calculate \( z^{10} \)
Since \( z = e^{i \frac{\pi}{10}} \), we have:

\[ z^{10} = \left(e^{i \frac{\pi}{10}}\right)^{10} = e^{i \pi} = \cos(\pi) + i \sin(\pi) = -1 + 0i = -1. \]

### Final Answer
Thus, the value of the given expression is

\[ \boxed{-1}. \]