If : `sin^(2) ((pi)/(4) + (theta)/(2)) - cos^(2) ((pi)/(4) + (theta)/(2))=` A)`sin theta` B)` cos theta` C)`sin 2 theta` D)`cos 2 theta`

Options

A. `sin theta`
B. ` cos theta`
C. `sin 2 theta`
D. `cos 2 theta`

Answer

To solve the equation \( \sin^2\left(\frac{\pi}{4} + \frac{\theta}{2}\right) - \cos^2\left(\frac{\pi}{4} + \frac{\theta}{2}\right) \), we can use trigonometric identities. Here are the steps:

### Step 1: Use the identity for sine and cosine squares
We know that:
\[ \sin^2 x - \cos^2 x = -\cos(2x) \]
Thus, we can rewrite our expression:
\[ \sin^2\left(\frac{\pi}{4} + \frac{\theta}{2}\right) - \cos^2\left(\frac{\pi}{4} + \frac{\theta}{2}\right) = -\cos\left(2\left(\frac{\pi}{4} + \frac{\theta}{2}\right)\right) \]

### Step 2: Simplify the argument of cosine
Now, simplify the argument of the cosine:
\[ 2\left(\frac{\pi}{4} + \frac{\theta}{2}\right) = \frac{\pi}{2} + \theta \]
So we have:
\[ -\cos\left(\frac{\pi}{2} + \theta\right) \]

### Step 3: Use the cosine addition formula
Using the cosine addition formula:
\[ \cos\left(\frac{\pi}{2} + \theta\right) = -\sin\theta \]
Thus, we can substitute this back into our expression:
\[ -\cos\left(\frac{\pi}{2} + \theta\right) = -(-\sin\theta) = \sin\theta \]

### Final Result
Therefore, the final result is:
\[ \sin^2\left(\frac{\pi}{4} + \frac{\theta}{2}\right) - \cos^2\left(\frac{\pi}{4} + \frac{\theta}{2}\right) = \sin\theta \]

### Answer
The correct option is A) \( \sin\theta \).