If `omega` is a cube root of unity, then `tan{(omega^(200)+(1)/(omega^(200)))pi+(pi)/(4)}=`

Options

A. 1
B. `(1)/(sqrt2)`
C. 0
D. none of these

Answer

To solve the problem, we need to find the value of

\[ \tan\left(\omega^{200} + \frac{1}{\omega^{200}}\right)\pi + \frac{\pi}{4} \]

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

### Step 1: Understanding Cube Roots of Unity
The cube roots of unity are the solutions to the equation \(x^3 = 1\). The roots are given by:
\[ \omega = e^{2\pi i / 3}, \quad \omega^2 = e^{4\pi i / 3}, \quad \text{and} \quad 1 \]
These satisfy the relations:
\[ 1 + \omega + \omega^2 = 0 \quad \text{and} \quad \omega^3 = 1 \]

### Step 2: Simplifying \(\omega^{200}\)
Since \(\omega^3 = 1\), we can reduce the exponent \(200\) modulo \(3\):
\[ 200 \mod 3 = 2 \]
Thus, we have:
\[ \omega^{200} = \omega^2 \]

### Step 3: Calculate \(\frac{1}{\omega^{200}}\)
Using the result from Step 2:
\[ \frac{1}{\omega^{200}} = \frac{1}{\omega^2} = \omega \]

### Step 4: Substitute into the Expression
Now substituting \(\omega^{200}\) and \(\frac{1}{\omega^{200}}\) into the expression:
\[ \tan\left(\omega^2 + \omega\right)\pi + \frac{\pi}{4} \]
From the relation \(1 + \omega + \omega^2 = 0\), we find:
\[ \omega + \omega^2 = -1 \]
Thus, we have:
\[ \tan(-\pi) + \frac{\pi}{4} \]

### Step 5: Simplifying the Tangent Function
Now we compute:
\[ \tan(-\pi) = 0 \]
So the expression simplifies to:
\[ \tan(0 + \frac{\pi}{4}) = \tan\left(\frac{\pi}{4}\right) \]

### Step 6: Final Calculation
We know:
\[ \tan\left(\frac{\pi}{4}\right) = 1 \]

### Conclusion
Thus, the final answer is:
\[ \boxed{1} \]