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 trigonometric identities for sine and cosine.

### Step-by-step Solution:

1. **Apply the sine addition formula**:
\[ \sin(45^\circ + \theta) = \sin 45^\circ \cos \theta + \cos 45^\circ \sin \theta \]
We know that \( \sin 45^\circ = \cos 45^\circ = \frac{1}{\sqrt{2}} \).
Therefore,
\[ \sin(45^\circ + \theta) = \frac{1}{\sqrt{2}} \cos \theta + \frac{1}{\sqrt{2}} \sin \theta \]
This simplifies to:
\[ \sin(45^\circ + \theta) = \frac{1}{\sqrt{2}} (\cos \theta + \sin \theta) \]

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

3. **Subtract the two results**:
Now we can substitute the results back into our original expression:
\[ \sin(45^\circ + \theta) - \cos(45^\circ - \theta) = \frac{1}{\sqrt{2}} (\cos \theta + \sin \theta) - \frac{1}{\sqrt{2}} (\cos \theta + \sin \theta) \]
This simplifies to:
\[ \sin(45^\circ + \theta) - \cos(45^\circ - \theta) = 0 \]

### Final Answer:
\[ \sin(45^\circ + \theta) - \cos(45^\circ - \theta) = 0 \]