The value of `cos^(2)((pi)/(4)+theta)-sin^(2)((pi)/(4)-theta)` is

Options

A. 0
B. `costheta`
C. `sin2theta`
D. `costheta`

Explanation

To solve the expression \( \cos^2\left(\frac{\pi}{4} + \theta\right) - \sin^2\left(\frac{\pi}{4} - \theta\right) \), we can utilize trigonometric identities.

### Step-by-Step Solution:

1. **Identify the angles**:
Let \( a = \frac{\pi}{4} + \theta \) and \( b = \frac{\pi}{4} - \theta \).

2. **Use the identity**:
We know that \( \cos^2 x - \sin^2 y = \cos^2 x + \sin^2 x - 2\sin^2 y \). However, a more useful identity here is:
\[ \cos^2 A - \sin^2 B = \cos(A + B)\cos(A - B) \]
In our case, we can rewrite the expression as:
\[ \cos^2\left(\frac{\pi}{4} + \theta\right) - \sin^2\left(\frac{\pi}{4} - \theta\right) \]

3. **Apply the identity**:
We can express this as:
\[ \cos^2 A - \sin^2 B = \cos\left(A + B\right)\cos\left(A - B\right) \]
where \( A = \frac{\pi}{4} + \theta \) and \( B = \frac{\pi}{4} - \theta \).

4. **Calculate \( A + B \) and \( A - B \)**:
\[ A + B = \left(\frac{\pi}{4} + \theta\right) + \left(\frac{\pi}{4} - \theta\right) = \frac{\pi}{2} \]
\[ A - B = \left(\frac{\pi}{4} + \theta\right) - \left(\frac{\pi}{4} - \theta\right) = 2\theta \]

5. **Substitute back into the identity**:
Now substituting these values into the identity gives:
\[ \cos^2\left(\frac{\pi}{4} + \theta\right) - \sin^2\left(\frac{\pi}{4} - \theta\right) = \cos\left(\frac{\pi}{2}\right) \cos(2\theta) \]

6. **Evaluate \( \cos\left(\frac{\pi}{2}\right) \)**:
We know that:
\[ \cos\left(\frac{\pi}{2}\right) = 0 \]
Thus, the entire expression simplifies to:
\[ 0 \cdot \cos(2\theta) = 0 \]

### Final Answer:
The value of \( \cos^2\left(\frac{\pi}{4} + \theta\right) - \sin^2\left(\frac{\pi}{4} - \theta\right) \) is \( 0 \).

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