If ` 0^(@) lt theta lt 90^(@) and cos^(2) theta - sin^(2) theta = (1)/(2)`, then what is the value of `theta` ?

Options

A. `30^(@)`
B. `45^(@)`
C. `60^(@)`
D. `90^(@)`

Answer

To solve the equation \( \cos^2 \theta - \sin^2 \theta = \frac{1}{2} \) for \( 0^\circ < \theta < 90^\circ \), we can follow these steps:

### Step 1: Use the Pythagorean Identity
We know from the Pythagorean identity that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
From this, we can express \( \sin^2 \theta \) in terms of \( \cos^2 \theta \):
\[ \sin^2 \theta = 1 - \cos^2 \theta \]

### Step 2: Substitute into the Given Equation
Now, substitute \( \sin^2 \theta \) into the equation:
\[ \cos^2 \theta - (1 - \cos^2 \theta) = \frac{1}{2} \]
This simplifies to:
\[ \cos^2 \theta - 1 + \cos^2 \theta = \frac{1}{2} \]
\[ 2\cos^2 \theta - 1 = \frac{1}{2} \]

### Step 3: Solve for \( \cos^2 \theta \)
Now, add 1 to both sides:
\[ 2\cos^2 \theta = \frac{1}{2} + 1 \]
\[ 2\cos^2 \theta = \frac{1}{2} + \frac{2}{2} = \frac{3}{2} \]
Now, divide both sides by 2:
\[ \cos^2 \theta = \frac{3}{4} \]

### Step 4: Take the Square Root
Taking the square root of both sides gives:
\[ \cos \theta = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2} \]

### Step 5: Find the Angle \( \theta \)
We know that \( \cos \theta = \frac{\sqrt{3}}{2} \) corresponds to:
\[ \theta = 30^\circ \]

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

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