The value of `2 sin^2 theta + k cos^2 2 theta =1 ` , then k is equal to :

Options

A. `cos theta + cos alpha`
B. independent of `theta`
C. independent of `alpha`
D. None of these

Answer

To solve the equation \( 2 \sin^2 \theta + k \cos^2 2\theta = 1 \) for \( k \), we can follow these steps:

### Step 1: Substitute a specific angle
Let's substitute \( \theta = \frac{\pi}{4} \) into the equation. This angle is chosen because \( \sin \frac{\pi}{4} = \cos \frac{\pi}{4} = \frac{1}{\sqrt{2}} \).

### Step 2: Calculate \( \sin^2 \theta \) and \( \cos^2 2\theta \)
Using \( \theta = \frac{\pi}{4} \):
- \( \sin^2 \frac{\pi}{4} = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \)
- Now, calculate \( \cos 2\theta \):
\[ \cos 2\theta = \cos \left(2 \cdot \frac{\pi}{4}\right) = \cos \frac{\pi}{2} = 0 \]
- Therefore, \( \cos^2 2\theta = 0^2 = 0 \).

### Step 3: Substitute values into the equation
Now substitute these values back into the original equation:
\[ 2 \sin^2 \frac{\pi}{4} + k \cos^2 2\frac{\pi}{4} = 1 \]
This simplifies to:
\[ 2 \cdot \frac{1}{2} + k \cdot 0 = 1 \]
\[ 1 + 0 = 1 \]

### Step 4: Analyze the equation
The equation \( 1 = 1 \) holds true for any value of \( k \). This indicates that \( k \) is independent of \( \theta \).

### Step 5: Conclusion
Since the equation holds true regardless of the value of \( k \), we conclude that \( k \) can take any value. However, we can express this as \( k \) being independent of \( \theta \).

Thus, the value of \( k \) is independent of \( \theta \).