Let `f(theta)=1/(1+(cottheta)^x)and S=sum_(theta=1^@)^(89^@) f(theta)`, then the value of `S` is _________ .

Explanation

To solve the problem, we need to evaluate the function \( f(\theta) = \frac{1}{1 + (\cot \theta)^x} \) and then find the sum \( S = \sum_{\theta=1}^{89} f(\theta) \).

### Step-by-Step Solution:

1. **Understanding the Function**:
The function is given as:
\[ f(\theta) = \frac{1}{1 + (\cot \theta)^x} \]
We can rewrite \( \cot \theta \) in terms of \( \tan \theta \):
\[ \cot \theta = \frac{1}{\tan \theta} \]
Thus, we have:
\[ f(\theta) = \frac{1}{1 + \left(\frac{1}{\tan \theta}\right)^x} = \frac{1}{1 + \frac{1}{(\tan \theta)^x}} = \frac{\tan^x \theta}{\tan^x \theta + 1} \]

2. **Finding \( f(90 - \theta) \)**:
Next, we evaluate \( f(90 - \theta) \):
\[ f(90 - \theta) = \frac{1}{1 + \cot(90 - \theta)^x} = \frac{1}{1 + \tan \theta^x} = \frac{1}{1 + (\tan \theta)^x} \]
This can also be rewritten as:
\[ f(90 - \theta) = \frac{1}{1 + \tan^x \theta} \]

3. **Adding \( f(\theta) \) and \( f(90 - \theta) \)**:
Now, we add \( f(\theta) \) and \( f(90 - \theta) \):
\[ f(\theta) + f(90 - \theta) = \frac{\tan^x \theta}{\tan^x \theta + 1} + \frac{1}{1 + \tan^x \theta} \]
Simplifying this:
\[ = \frac{\tan^x \theta + 1}{\tan^x \theta + 1} = 1 \]
This shows that \( f(\theta) + f(90 - \theta) = 1 \).

4. **Summing \( S \)**:
The sum \( S \) can be expressed as:
\[ S = \sum_{\theta=1}^{89} f(\theta) \]
We can pair the terms:
\[ S = (f(1) + f(89)) + (f(2) + f(88)) + \ldots + (f(44) + f(46)) + f(45) \]
Each pair \( f(\theta) + f(90 - \theta) = 1 \) contributes 1 to the sum. There are 44 such pairs from \( \theta = 1 \) to \( 44 \).

5. **Calculating \( f(45) \)**:
Now we need to calculate \( f(45) \):
\[ f(45) = \frac{1}{1 + \cot(45)^x} = \frac{1}{1 + 1^x} = \frac{1}{1 + 1} = \frac{1}{2} \]

6. **Final Calculation**:
Thus, the total sum \( S \) is:
\[ S = 44 \cdot 1 + \frac{1}{2} = 44 + 0.5 = 44.5 \]

### Final Answer:
The value of \( S \) is \( \boxed{44.5} \).