If `8cos2theta+8sec2theta=65and0^(@)ltthetalt(pi)/(2)`, then `4cos4theta` is equal to

Options

A. a) `-(23)/(8)`
B. b) `-(31)/(8)`
C. c) `-(31)/(32)`
D. d) `-(33)/(32)`

Answer

To solve the problem \(8 \cos 2\theta + 8 \sec 2\theta = 65\) and find \(4 \cos 4\theta\), we can follow these steps:

### Step 1: Rewrite the equation
Given:
\[ 8 \cos 2\theta + 8 \sec 2\theta = 65 \]
We can rewrite \(\sec 2\theta\) as \(\frac{1}{\cos 2\theta}\):
\[ 8 \cos 2\theta + \frac{8}{\cos 2\theta} = 65 \]

### Step 2: Multiply through by \(\cos 2\theta\)
To eliminate the fraction, multiply both sides by \(\cos 2\theta\):
\[ 8 \cos^2 2\theta + 8 = 65 \cos 2\theta \]

### Step 3: Rearrange the equation
Rearranging gives us:
\[ 8 \cos^2 2\theta - 65 \cos 2\theta + 8 = 0 \]

### Step 4: Let \(x = \cos 2\theta\)
Substituting \(x\) for \(\cos 2\theta\) gives us a quadratic equation:
\[ 8x^2 - 65x + 8 = 0 \]

### Step 5: Use the quadratic formula
Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
Here, \(a = 8\), \(b = -65\), and \(c = 8\):
\[ x = \frac{65 \pm \sqrt{(-65)^2 - 4 \cdot 8 \cdot 8}}{2 \cdot 8} \]
Calculating the discriminant:
\[ (-65)^2 - 4 \cdot 8 \cdot 8 = 4225 - 256 = 3969 \]
Thus,
\[ x = \frac{65 \pm \sqrt{3969}}{16} \]
Since \(\sqrt{3969} = 63\):
\[ x = \frac{65 \pm 63}{16} \]

### Step 6: Solve for \(x\)
Calculating the two possible values for \(x\):
1. \(x = \frac{128}{16} = 8\) (not possible since \(\cos\) cannot be greater than 1)
2. \(x = \frac{2}{16} = \frac{1}{8}\)

Thus, we have:
\[ \cos 2\theta = \frac{1}{8} \]

### Step 7: Find \(\cos 4\theta\)
Using the double angle formula:
\[ \cos 4\theta = 2 \cos^2 2\theta - 1 \]
Substituting \(\cos 2\theta = \frac{1}{8}\):
\[ \cos 4\theta = 2 \left(\frac{1}{8}\right)^2 - 1 = 2 \cdot \frac{1}{64} - 1 = \frac{2}{64} - 1 = \frac{1}{32} - 1 = \frac{1 - 32}{32} = \frac{-31}{32} \]

### Step 8: Calculate \(4 \cos 4\theta\)
Now, we multiply by 4:
\[ 4 \cos 4\theta = 4 \cdot \frac{-31}{32} = \frac{-124}{32} = \frac{-31}{8} \]

Thus, the final answer is:
\[ \boxed{\frac{-31}{8}} \]