Find the value of `(i^(592)+i^(590)+i^(588)+i^(586)+i^(584))/(i^(582)+i^(580)+i^(578)+i^(576)+i^(574))-1` `(1+i)^6+(1-i)^6`

Options

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

Explanation

To solve the expression

\[ \frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}} - 1, \]

we will first simplify the numerator and the denominator.

### Step 1: Factor out the lowest power in the numerator

The lowest power in the numerator is \(i^{584}\). We can factor this out:

\[ i^{584}(i^{8} + i^{6} + i^{4} + i^{2} + 1). \]

### Step 2: Factor out the lowest power in the denominator

The lowest power in the denominator is \(i^{574}\). We can factor this out:

\[ i^{574}(i^{8} + i^{6} + i^{4} + i^{2} + 1). \]

### Step 3: Substitute back into the expression

Now, substituting these factored forms back into the expression gives us:

\[ \frac{i^{584}(i^{8} + i^{6} + i^{4} + i^{2} + 1)}{i^{574}(i^{8} + i^{6} + i^{4} + i^{2} + 1)} - 1. \]

### Step 4: Cancel the common terms

The common term \(i^{8} + i^{6} + i^{4} + i^{2} + 1\) cancels out:

\[ \frac{i^{584}}{i^{574}} - 1. \]

### Step 5: Simplify the fraction

This simplifies to:

\[ i^{10} - 1. \]

### Step 6: Calculate \(i^{10}\)

We know that \(i^2 = -1\), so we can find \(i^{10}\):

\[ i^{10} = (i^2)^5 = (-1)^5 = -1. \]

### Step 7: Substitute back into the expression

Now substituting back, we have:

\[ -1 - 1 = -2. \]

### Final Answer

Thus, the value of the expression is:

\[ \boxed{-2}. \]

---