If `tantheta+cottheta=x` , then what is the value of `tan^(4)theta+cot^(4)theta`?

Options

A. `(x^(3)-3)^(2)+2`
B. `(x^(4)-2x)+4`
C. `x(x-4)+2`
D. `x^(2)(x^(2)-4)+2`

Answer

To solve the problem, we need to find the value of \( \tan^4 \theta + \cot^4 \theta \) given that \( \tan \theta + \cot \theta = x \).

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ \tan \theta + \cot \theta = x \]

2. **Square both sides:**
\[ (\tan \theta + \cot \theta)^2 = x^2 \]
Expanding the left side:
\[ \tan^2 \theta + 2 \tan \theta \cot \theta + \cot^2 \theta = x^2 \]
Since \( \tan \theta \cot \theta = 1 \):
\[ \tan^2 \theta + 2 + \cot^2 \theta = x^2 \]
Rearranging gives:
\[ \tan^2 \theta + \cot^2 \theta = x^2 - 2 \]

3. **Now, square the result from step 2:**
\[ (\tan^2 \theta + \cot^2 \theta)^2 = (x^2 - 2)^2 \]
Expanding the left side:
\[ \tan^4 \theta + 2 \tan^2 \theta \cot^2 \theta + \cot^4 \theta = (x^2 - 2)^2 \]
Again, since \( \tan^2 \theta \cot^2 \theta = 1 \):
\[ \tan^4 \theta + 2 + \cot^4 \theta = (x^2 - 2)^2 \]
Rearranging gives:
\[ \tan^4 \theta + \cot^4 \theta = (x^2 - 2)^2 - 2 \]

4. **Expand the right side:**
\[ (x^2 - 2)^2 = x^4 - 4x^2 + 4 \]
Therefore:
\[ \tan^4 \theta + \cot^4 \theta = x^4 - 4x^2 + 4 - 2 \]
Simplifying this:
\[ \tan^4 \theta + \cot^4 \theta = x^4 - 4x^2 + 2 \]

5. **Final expression:**
We can factor this expression:
\[ \tan^4 \theta + \cot^4 \theta = (x^2 - 4)x^2 + 2 \]

### Conclusion:
Thus, the value of \( \tan^4 \theta + \cot^4 \theta \) is:
\[ \tan^4 \theta + \cot^4 \theta = x^4 - 4x^2 + 2 \]