The equation `2("cos"^(8)theta - "sin"^(8)theta) "sec" 2 theta =a^(2)` has real solution if a lies in the interval

Options

A. `[-sqrt(2), sqrt(2)]`
B. `[-sqrt(2), -1) cup (1, sqrt(2)]`
C. `[-sqrt(2), -1] cup [1, sqrt(2)]`
D. none of these

Explanation

To solve the equation \( 2(\cos^8 \theta - \sin^8 \theta) \sec 2\theta = a^2 \) for the values of \( a \) that allow for real solutions, we will follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ a^2 = 2(\cos^8 \theta - \sin^8 \theta) \sec 2\theta \]
Using the identity \( \cos^8 \theta - \sin^8 \theta = (\cos^4 \theta - \sin^4 \theta)(\cos^4 \theta + \sin^4 \theta) \) and the difference of squares, we can rewrite:
\[ \cos^8 \theta - \sin^8 \theta = (\cos^4 \theta - \sin^4 \theta)(\cos^4 \theta + \sin^4 \theta) \]
This can further be expressed as:
\[ \cos^4 \theta - \sin^4 \theta = (\cos^2 \theta - \sin^2 \theta)(\cos^2 \theta + \sin^2 \theta) = (\cos^2 \theta - \sin^2 \theta) \]
since \( \cos^2 \theta + \sin^2 \theta = 1 \).

### Step 2: Substitute back into the equation
Thus, we can write:
\[ a^2 = 2(\cos^4 \theta - \sin^4 \theta) \sec 2\theta = 2(\cos^2 \theta - \sin^2 \theta) \sec 2\theta \]

### Step 3: Use the identity for secant
Recall that \( \sec 2\theta = \frac{1}{\cos 2\theta} \), so we can substitute:
\[ a^2 = \frac{2(\cos^2 \theta - \sin^2 \theta)}{\cos 2\theta} \]

### Step 4: Simplify using trigonometric identities
Using the identity \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \), we can express:
\[ a^2 = \frac{2(\cos^2 \theta - \sin^2 \theta)}{\cos^2 \theta - \sin^2 \theta} = 2 \]
This means \( a^2 \) can take the value of 2.

### Step 5: Determine the range of \( a \)
Since \( a^2 = 2 \), we find that \( a \) can be:
\[ a = \pm \sqrt{2} \]

### Step 6: Conclude the interval for \( a \)
Thus, the values of \( a \) that allow for real solutions are:
\[ a \in [-\sqrt{2}, \sqrt{2}] \]

### Final Answer
The equation \( 2(\cos^8 \theta - \sin^8 \theta) \sec 2\theta = a^2 \) has real solutions if \( a \) lies in the interval:
\[ [-\sqrt{2}, \sqrt{2}] \]

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