Let `x` be the solution set of equation `A^x=Idot,w h e r eA+[0 1-1 4-3 4 3-3 4]a n dI` is the corresponding unit matrix and `xsubeN ,` then the minimum value of `sum(cos^xtheta+sin^xtheta),theta in Rdot`

Explanation

To solve the problem, we need to find the minimum value of the expression \( \sum ( \cos^x \theta + \sin^x \theta ) \) given that \( A^x = I \) for a matrix \( A \) and \( x \in \mathbb{N} \).

### Step-by-Step Solution:

1. **Identify the Matrix \( A \)**:
We have the matrix:
\[ A = \begin{pmatrix} 0 & 1 \\ -1 & 4 \\ -3 & 4 \\ 3 & -3 \\ 4 \end{pmatrix} \]

2. **Calculate \( A^2 \)**:
We need to find \( A^2 \) to check if it equals the identity matrix \( I \).
\[ A^2 = A \times A \]
Performing the multiplication:
\[ A^2 = \begin{pmatrix} 0 & 1 \\ -1 & 4 \\ -3 & 4 \\ 3 & -3 \\ 4 \end{pmatrix} \times \begin{pmatrix} 0 & 1 \\ -1 & 4 \\ -3 & 4 \\ 3 & -3 \\ 4 \end{pmatrix} \]
After performing the matrix multiplication, we find:
\[ A^2 = I \]
where \( I \) is the identity matrix.

3. **Determine Values of \( x \)**:
Since \( A^2 = I \), we can conclude that:
\[ A^x = I \text{ for even } x \text{ (i.e., } x = 2, 4, 6, \ldots \text{)} \]
Thus, \( x \) can take values from the set of natural numbers \( \mathbb{N} \) that are even.

4. **Evaluate the Expression**:
We need to evaluate:
\[ \sum ( \cos^x \theta + \sin^x \theta ) \]
for even values of \( x \):
\[ \sum_{k=1}^{n} ( \cos^{2k} \theta + \sin^{2k} \theta ) \]
This can be separated into two geometric series.

5. **Sum of Geometric Series**:
The sum of the series for \( \cos^{2k} \theta \) and \( \sin^{2k} \theta \):
\[ S_{\cos} = \sum_{k=1}^{n} \cos^{2k} \theta = \frac{\cos^2 \theta}{1 - \cos^2 \theta} \]
\[ S_{\sin} = \sum_{k=1}^{n} \sin^{2k} \theta = \frac{\sin^2 \theta}{1 - \sin^2 \theta} \]

6. **Combine the Sums**:
Combining these results:
\[ S = S_{\cos} + S_{\sin} = \frac{\cos^2 \theta}{\sin^2 \theta} + \frac{\sin^2 \theta}{\cos^2 \theta} \]
Let \( u = \tan^2 \theta \):
\[ S = u + \frac{1}{u} \]

7. **Find the Minimum Value**:
To find the minimum value of \( u + \frac{1}{u} \), we apply the AM-GM inequality:
\[ u + \frac{1}{u} \geq 2 \]
The minimum value occurs when \( u = 1 \) (i.e., \( \tan^2 \theta = 1 \), or \( \theta = \frac{\pi}{4} \)).

### Final Answer:
The minimum value of \( \sum ( \cos^x \theta + \sin^x \theta ) \) is:
\[ \boxed{2} \]