The value of `sin(45^(@)+theta)-cos(45^(@)-theta)` is

Options

A. `2costheta`
B. `2sintheta`
C. 1
D. 0

Explanation

To find the value of \( \sin(45^\circ + \theta) - \cos(45^\circ - \theta) \), we can use the sine and cosine addition formulas.

### Step-by-Step Solution:

1. **Use the sine addition formula**:
\[ \sin(45^\circ + \theta) = \sin 45^\circ \cos \theta + \cos 45^\circ \sin \theta \]
We know that \( \sin 45^\circ = \frac{1}{\sqrt{2}} \) and \( \cos 45^\circ = \frac{1}{\sqrt{2}} \). Therefore, we can substitute these values:
\[ \sin(45^\circ + \theta) = \frac{1}{\sqrt{2}} \cos \theta + \frac{1}{\sqrt{2}} \sin \theta \]

2. **Use the cosine subtraction formula**:
\[ \cos(45^\circ - \theta) = \cos 45^\circ \cos \theta + \sin 45^\circ \sin \theta \]
Again substituting \( \cos 45^\circ = \frac{1}{\sqrt{2}} \) and \( \sin 45^\circ = \frac{1}{\sqrt{2}} \):
\[ \cos(45^\circ - \theta) = \frac{1}{\sqrt{2}} \cos \theta + \frac{1}{\sqrt{2}} \sin \theta \]

3. **Combine the results**:
Now we can substitute these results back into the original expression:
\[ \sin(45^\circ + \theta) - \cos(45^\circ - \theta) = \left(\frac{1}{\sqrt{2}} \cos \theta + \frac{1}{\sqrt{2}} \sin \theta\right) - \left(\frac{1}{\sqrt{2}} \cos \theta + \frac{1}{\sqrt{2}} \sin \theta\right) \]

4. **Simplify the expression**:
\[ = \frac{1}{\sqrt{2}} \cos \theta + \frac{1}{\sqrt{2}} \sin \theta - \frac{1}{\sqrt{2}} \cos \theta - \frac{1}{\sqrt{2}} \sin \theta \]
All terms cancel out:
\[ = 0 \]

### Final Answer:
The value of \( \sin(45^\circ + \theta) - \cos(45^\circ - \theta) \) is \( 0 \).