If `2(cos^(2)theta-sin^(2)theta)=1,theta` is a positive acute angle , then the value of `theta` is

Options

A. `60^(@)`
B. `30^(@)`
C. `45^(@)`
D. `22(1)/(2)@`

Answer

To solve the equation \( 2(\cos^2 \theta - \sin^2 \theta) = 1 \) for the positive acute angle \( \theta \), we can follow these steps:

### Step 1: Rewrite the equation
Start with the given equation:
\[ 2(\cos^2 \theta - \sin^2 \theta) = 1 \]

### Step 2: Simplify the equation
Divide both sides by 2:
\[ \cos^2 \theta - \sin^2 \theta = \frac{1}{2} \]

### Step 3: Use the double angle identity
Recall the double angle identity for cosine:
\[ \cos 2\theta = \cos^2 \theta - \sin^2 \theta \]
So we can rewrite the equation as:
\[ \cos 2\theta = \frac{1}{2} \]

### Step 4: Find the angle for cosine
We know that \( \cos 60^\circ = \frac{1}{2} \). Therefore, we can set up the equation:
\[ 2\theta = 60^\circ \]

### Step 5: Solve for \( \theta \)
Now, divide both sides by 2 to find \( \theta \):
\[ \theta = \frac{60^\circ}{2} = 30^\circ \]

### Conclusion
Thus, the value of \( \theta \) is:
\[ \theta = 30^\circ \]