The sum of all values of `theta in (0,pi/2)` satisfying `sin^(2)2theta+cos^(4)2theta=3/4` is

Options

A. `pi/2`
B. `pi`
C. `(3pi)/8`
D. `(5pi)/4`

Answer

To solve the equation \( \sin^2(2\theta) + \cos^4(2\theta) = \frac{3}{4} \) for \( \theta \) in the interval \( (0, \frac{\pi}{2}) \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ \sin^2(2\theta) + \cos^4(2\theta) = \frac{3}{4} \]
We know that \( \cos^4(2\theta) = (\cos^2(2\theta))^2 \). Let \( x = \cos^2(2\theta) \). Then, \( \sin^2(2\theta) = 1 - x \).

Substituting this into the equation gives:
\[ 1 - x + x^2 = \frac{3}{4} \]

### Step 2: Rearranging the equation
Rearranging the equation, we have:
\[ x^2 - x + 1 - \frac{3}{4} = 0 \]
This simplifies to:
\[ x^2 - x + \frac{1}{4} = 0 \]

### Step 3: Solving the quadratic equation
Now, we can solve the quadratic equation \( x^2 - x + \frac{1}{4} = 0 \) using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1, b = -1, c = \frac{1}{4} \).

Calculating the discriminant:
\[ b^2 - 4ac = (-1)^2 - 4 \cdot 1 \cdot \frac{1}{4} = 1 - 1 = 0 \]
Since the discriminant is 0, there is one real solution:
\[ x = \frac{1 \pm 0}{2} = \frac{1}{2} \]

### Step 4: Finding \( \cos^2(2\theta) \)
Since \( x = \cos^2(2\theta) \), we have:
\[ \cos^2(2\theta) = \frac{1}{2} \]
Taking the square root gives:
\[ \cos(2\theta) = \frac{1}{\sqrt{2}} \quad \text{or} \quad \cos(2\theta) = -\frac{1}{\sqrt{2}} \]
However, since \( \theta \) is in \( (0, \frac{\pi}{2}) \), we only consider:
\[ \cos(2\theta) = \frac{1}{\sqrt{2}} \]

### Step 5: Finding \( 2\theta \)
The solution for \( \cos(2\theta) = \frac{1}{\sqrt{2}} \) gives:
\[ 2\theta = \frac{\pi}{4} + 2k\pi \quad \text{or} \quad 2\theta = -\frac{\pi}{4} + 2k\pi \]
For \( k = 0 \):
1. \( 2\theta = \frac{\pi}{4} \) implies \( \theta = \frac{\pi}{8} \)
2. \( 2\theta = -\frac{\pi}{4} \) is not in the range since \( \theta \) must be positive.

### Step 6: Conclusion
Thus, the only solution for \( \theta \) in \( (0, \frac{\pi}{2}) \) is:
\[ \theta = \frac{\pi}{8} \]

### Final Answer
The sum of all values of \( \theta \) satisfying the equation in the interval \( (0, \frac{\pi}{2}) \) is:
\[ \frac{\pi}{8} \]