If `omega` is a complex cube root of unity, then value of expression `cos [{(1-omega)(1-omega^(2))+...+(12-omega)(12-omega^(2))}(pi)/(370)]`

Options

A. -1
B. 0
C. 1
D. `sqrt(3)//2`

Answer

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

\[ \cos \left( \frac{(1 - \omega)(1 - \omega^2) + (2 - \omega)(2 - \omega^2) + \ldots + (12 - \omega)(12 - \omega^2)}{370} \cdot \pi \right) \]

where \(\omega\) is a complex cube root of unity.

### Step 1: Simplify the expression inside the cosine

We start with the term \((a - \omega)(a - \omega^2)\) for \(a = 1, 2, \ldots, 12\):

\[ (a - \omega)(a - \omega^2) = a^2 - a(\omega + \omega^2) + \omega \cdot \omega^2 \]

Since \(\omega^3 = 1\), we have \(\omega \cdot \omega^2 = \omega^3 = 1\). Also, from the properties of cube roots of unity, we know that:

\[ 1 + \omega + \omega^2 = 0 \implies \omega + \omega^2 = -1 \]

Thus, we can rewrite the expression as:

\[ (a - \omega)(a - \omega^2) = a^2 + a - 1 \]

### Step 2: Sum the expression from \(a = 1\) to \(12\)

Now, we need to sum this from \(a = 1\) to \(12\):

\[ \sum_{a=1}^{12} \left( a^2 + a - 1 \right) = \sum_{a=1}^{12} a^2 + \sum_{a=1}^{12} a - \sum_{a=1}^{12} 1 \]

### Step 3: Calculate each summation

1. **Sum of squares**: The formula for the sum of squares of the first \(n\) natural numbers is:

\[ \sum_{a=1}^{n} a^2 = \frac{n(n + 1)(2n + 1)}{6} \]

For \(n = 12\):

\[ \sum_{a=1}^{12} a^2 = \frac{12 \cdot 13 \cdot 25}{6} = 650 \]

2. **Sum of first \(n\) natural numbers**: The formula is:

\[ \sum_{a=1}^{n} a = \frac{n(n + 1)}{2} \]

For \(n = 12\):

\[ \sum_{a=1}^{12} a = \frac{12 \cdot 13}{2} = 78 \]

3. **Sum of \(1\)**: This is simply \(12\) since we are summing \(1\) twelve times.

### Step 4: Combine the results

Now we can combine these results:

\[ \sum_{a=1}^{12} (a^2 + a - 1) = 650 + 78 - 12 = 716 \]

### Step 5: Substitute back into the cosine expression

Now we substitute back into the cosine expression:

\[ \cos \left( \frac{716 \pi}{370} \right) \]

### Step 6: Simplify the fraction

We can simplify \(\frac{716}{370}\):

\[ \frac{716}{370} = \frac{358}{185} \quad \text{(dividing both by 2)} \]

### Step 7: Evaluate the cosine

Now we need to evaluate:

\[ \cos \left( \frac{358 \pi}{185} \right) \]

Since \(358\) is \(2 \cdot 185 - 12\), we can reduce this further:

\[ \cos \left( 2\pi - \frac{12\pi}{185} \right) = \cos \left( -\frac{12\pi}{185} \right) = \cos \left( \frac{12\pi}{185} \right) \]

### Final Answer

Thus, the final answer is:

\[ \cos(2\pi) = 1 \]