If the sum of all values of ` theta , 0 le theta le 2 pi ` satisfying the equation ` (8 cos 4 theta - 3) ( cot theta + tan theta -2 ) ( cot theta + tan theta + 2) = 12 ` is ` k pi `, then k is equal to :

Options

A. 2
B. 4
C. `12`
D. `8`

Answer

To solve the equation \((8 \cos 4\theta - 3)(\cot \theta + \tan \theta - 2)(\cot \theta + \tan \theta + 2) = 12\) for \(0 \leq \theta \leq 2\pi\), we will follow these steps:

### Step 1: Rewrite the Equation
We can rewrite the equation as:
\[ (8 \cos 4\theta - 3) \left( (\cot \theta + \tan \theta)^2 - 4 \right) = 12 \]
This is because \((a + b)(a - b) = a^2 - b^2\).

### Step 2: Simplify the Expression
Let \(x = \cot \theta + \tan \theta\). Then the equation becomes:
\[ (8 \cos 4\theta - 3)(x^2 - 4) = 12 \]

### Step 3: Expand and Rearrange
Expanding the equation gives:
\[ 8 \cos 4\theta x^2 - 32 \cos 4\theta - 3x^2 + 12 = 0 \]
This can be rearranged to:
\[ 8 \cos 4\theta x^2 - 3x^2 - 32 \cos 4\theta + 12 = 0 \]

### Step 4: Factor Out Common Terms
We can factor out common terms:
\[ (8 \cos 4\theta - 3)x^2 + 12 - 32 \cos 4\theta = 0 \]

### Step 5: Solve for \(x\)
Now we will solve for \(x\):
\[ x^2 = \frac{32 \cos 4\theta - 12}{8 \cos 4\theta - 3} \]

### Step 6: Find Values of \(\theta\)
Next, we need to find the values of \(\theta\) that satisfy this equation. We know that:
\[ \cot \theta + \tan \theta = \frac{\cos^2 \theta + \sin^2 \theta}{\sin \theta \cos \theta} = \frac{1}{\sin \theta \cos \theta} = \frac{2}{\sin 2\theta} \]
Thus:
\[ \frac{2}{\sin 2\theta} = x \]

### Step 7: Substitute and Solve
Substituting back into our equation, we can find the angles that satisfy:
\[ \sin 2\theta = \frac{2}{x} \]

### Step 8: Identify Possible Values of \(\theta\)
From the values of \(\cos 4\theta\), we can find:
\[ \cos 4\theta = \frac{3}{8} \quad \text{or} \quad \cos 4\theta = \frac{11}{16} \]

### Step 9: Calculate the Angles
Using the inverse cosine function, we can determine:
\[ 4\theta = \cos^{-1}\left(\frac{3}{8}\right), \quad 4\theta = \cos^{-1}\left(\frac{11}{16}\right) \]
This gives us multiple angles for \(\theta\).

### Step 10: Sum the Values of \(\theta\)
After calculating all possible angles for \(\theta\) in the interval \([0, 2\pi]\), we sum these values.

### Final Step: Calculate \(k\)
The sum of all values of \(\theta\) is given as \(k\pi\). After performing the calculations, we find that:
\[ k = 8 \]

Thus, the final answer is:
\[ \boxed{8} \]