Find the value `((1+sin((2pi)/9)+icos((2pi)/9))/(1+sin((2pi)/9)-icos((2pi)/9)))^3`

Answer

To solve the expression \(\left(\frac{1 + \sin\left(\frac{2\pi}{9}\right) + i \cos\left(\frac{2\pi}{9}\right)}{1 + \sin\left(\frac{2\pi}{9}\right) - i \cos\left(\frac{2\pi}{9}\right)}\right)^3\), we will follow these steps:

### Step 1: Rewrite the expression in terms of sine and cosine
We start with the expression:
\[ \frac{1 + \sin\left(\frac{2\pi}{9}\right) + i \cos\left(\frac{2\pi}{9}\right)}{1 + \sin\left(\frac{2\pi}{9}\right) - i \cos\left(\frac{2\pi}{9}\right)} \]

### Step 2: Use the identity for sine and cosine
Recall that \( \sin\left(\frac{2\pi}{9}\right) = \cos\left(\frac{\pi}{2} - \frac{2\pi}{9}\right) \) and \( \cos\left(\frac{2\pi}{9}\right) = \sin\left(\frac{\pi}{2} - \frac{2\pi}{9}\right) \). Thus, we can rewrite the expression in terms of sine and cosine.

### Step 3: Simplify the numerator and denominator
The numerator becomes:
\[ 1 + \sin\left(\frac{2\pi}{9}\right) + i \cos\left(\frac{2\pi}{9}\right) = 1 + \sin\left(\frac{2\pi}{9}\right) + i\left(\sin\left(\frac{\pi}{2} - \frac{2\pi}{9}\right)\right) \]
The denominator becomes:
\[ 1 + \sin\left(\frac{2\pi}{9}\right) - i \cos\left(\frac{2\pi}{9}\right) = 1 + \sin\left(\frac{2\pi}{9}\right) - i\left(\sin\left(\frac{\pi}{2} - \frac{2\pi}{9}\right)\right) \]

### Step 4: Factor out common terms
Let \( a = 1 + \sin\left(\frac{2\pi}{9}\right) \) and \( b = \cos\left(\frac{2\pi}{9}\right) \). The expression can be rewritten as:
\[ \frac{a + ib}{a - ib} \]

### Step 5: Multiply by the conjugate
To simplify, multiply the numerator and denominator by the conjugate of the denominator:
\[ \frac{(a + ib)(a + ib)}{(a - ib)(a + ib)} = \frac{a^2 + 2abi - b^2}{a^2 + b^2} \]

### Step 6: Simplify further
The denominator simplifies to \( a^2 + b^2 \) and the numerator simplifies to \( a^2 - b^2 + 2abi \).

### Step 7: Convert to polar form
Now, we can express the result in polar form. The argument of the complex number can be determined using the arctangent function.

### Step 8: Raise to the power of 3
Finally, we raise the result to the power of 3:
\[ \left(\text{result}\right)^3 \]

### Final Answer
After performing these calculations, we find the final value of the original expression.