Let `A=x^(4)+4x^(3)+2x^(2)-4x+7` where `x=cot.(11pi)/(8) and B=(1-cos 8theta)/(tan^(2)4theta)+(1+cos 8theta)/(cot^(2)4theta)` where `theta=9^(@)`, then the value of `(AxxB)/(2)` is equal to

Answer

To solve the problem, we need to evaluate the expressions for \( A \) and \( B \) given the values of \( x \) and \( \theta \), and then find the value of \( \frac{A \times B}{2} \).

### Step 1: Calculate \( A \)

Given:
\[ A = x^4 + 4x^3 + 2x^2 - 4x + 7 \]
where \( x = \cot\left(\frac{11\pi}{8}\right) \).

First, we simplify \( x \):
\[ \cot\left(\frac{11\pi}{8}\right) = \cot\left(\pi + \frac{3\pi}{8}\right) = \cot\left(\frac{3\pi}{8}\right) \]
Since \( \cot(\pi + \theta) = \cot(\theta) \), we find:
\[ x = \cot\left(\frac{3\pi}{8}\right) \]
Now, using the identity \( \cot\left(\frac{\pi}{2} - \theta\right) = \tan(\theta) \):
\[ x = \tan\left(\frac{\pi}{8}\right) \]

Next, we need to calculate \( A \) using \( x = \tan\left(\frac{\pi}{8}\right) \). We know that \( \tan\left(\frac{\pi}{8}\right) \) can be expressed as \( x \).

To find \( A \), we can substitute \( x \) into the polynomial:
1. Calculate \( x^2 = \tan^2\left(\frac{\pi}{8}\right) \).
2. Calculate \( x^3 = \tan^3\left(\frac{\pi}{8}\right) \).
3. Calculate \( x^4 = \tan^4\left(\frac{\pi}{8}\right) \).

However, we can also use the result of \( x^2 + 2x - 1 = 0 \) derived from the quadratic equation to find \( A \) directly.

### Step 2: Calculate \( B \)

Given:
\[ B = \frac{1 - \cos(8\theta)}{\tan^2(4\theta)} + \frac{1 + \cos(8\theta)}{\cot^2(4\theta)} \]
where \( \theta = 9^\circ \).

Using the double angle formulas:
\[ \cos(8\theta) = 2\cos^2(4\theta) - 1 \]
Substituting this into \( B \):
\[ B = \frac{1 - (2\cos^2(4\theta) - 1)}{\tan^2(4\theta)} + \frac{1 + (2\cos^2(4\theta) - 1)}{\cot^2(4\theta)} \]
Simplifying:
\[ B = \frac{2 - 2\cos^2(4\theta)}{\tan^2(4\theta)} + \frac{2\cos^2(4\theta)}{\cot^2(4\theta)} \]
This can be simplified further to:
\[ B = 2\left(\frac{1 - \cos^2(4\theta)}{\tan^2(4\theta)} + \frac{\cos^2(4\theta)}{\cot^2(4\theta)}\right) \]
Using \( \sin^2(4\theta) + \cos^2(4\theta) = 1 \):
\[ B = 2 \]

### Step 3: Calculate \( \frac{A \times B}{2} \)

Now we have:
- \( A = 6 \)
- \( B = 2 \)

Calculating:
\[ \frac{A \times B}{2} = \frac{6 \times 2}{2} = 6 \]

### Final Answer:
\[ \frac{A \times B}{2} = 6 \]