Let `f (theta)=(1)/(1+(tan theta)^(2021))`, then the value of `sum_(theta=1^(@))^(89^(@))f(theta)` is equal to

Options

A. 45
B. 44
C. `(89)/(2)`
D. `(91)/(2)`

Answer

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

\[ S = \sum_{\theta=1}^{89} f(\theta) = \sum_{\theta=1}^{89} \frac{1}{1 + (\tan \theta)^{2021}} \]

### Step 1: Understanding the function
We start with the function:

\[ f(\theta) = \frac{1}{1 + (\tan \theta)^{2021}} \]

### Step 2: Using the identity for tangent
We know the identity:

\[ \tan(90^\circ - \theta) = \cot(\theta) \]

This means that:

\[ f(90^\circ - \theta) = \frac{1}{1 + (\tan(90^\circ - \theta))^{2021}} = \frac{1}{1 + (\cot \theta)^{2021}} \]

### Step 3: Simplifying \( f(90^\circ - \theta) \)
Now, we can express \( f(90^\circ - \theta) \):

\[ f(90^\circ - \theta) = \frac{1}{1 + \frac{1}{(\tan \theta)^{2021}}} = \frac{(\tan \theta)^{2021}}{(\tan \theta)^{2021} + 1} \]

### Step 4: Adding \( f(\theta) \) and \( f(90^\circ - \theta) \)
Now, we can add \( f(\theta) \) and \( f(90^\circ - \theta) \):

\[ f(\theta) + f(90^\circ - \theta) = \frac{1}{1 + (\tan \theta)^{2021}} + \frac{(\tan \theta)^{2021}}{(\tan \theta)^{2021} + 1} \]

### Step 5: Finding a common denominator
The common denominator is \( (1 + (\tan \theta)^{2021}) \):

\[ = \frac{1 + (\tan \theta)^{2021}}{1 + (\tan \theta)^{2021}} = 1 \]

Thus, we have:

\[ f(\theta) + f(90^\circ - \theta) = 1 \]

### Step 6: Pairing terms in the sum
Now, we can pair the terms in the sum \( S \):

\[ S = \sum_{\theta=1}^{44} \left( f(\theta) + f(90^\circ - \theta) \right) + f(45^\circ) \]

### Step 7: Evaluating the pairs
Since \( f(\theta) + f(90^\circ - \theta) = 1 \), we have:

\[ S = \sum_{\theta=1}^{44} 1 + f(45^\circ) \]

### Step 8: Evaluating \( f(45^\circ) \)
Now, we calculate \( f(45^\circ) \):

\[ f(45^\circ) = \frac{1}{1 + (\tan 45^\circ)^{2021}} = \frac{1}{1 + 1^{2021}} = \frac{1}{2} \]

### Step 9: Completing the sum
Now we can complete the sum:

\[ S = 44 + \frac{1}{2} = 44.5 = \frac{89}{2} \]

### Final Answer
Thus, the value of the sum is:

\[ \boxed{\frac{89}{2}} \]