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

Options

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

Answer

To solve the problem, we need to find the value of the expression:

\[ \sin(45^\circ + \theta) - \cos(45^\circ - \theta) \]

### Step 1: Apply the Sine Addition Formula
Using the sine addition formula, we have:

\[ \sin(A + B) = \sin A \cos B + \cos A \sin B \]

For \(A = 45^\circ\) and \(B = \theta\):

\[ \sin(45^\circ + \theta) = \sin 45^\circ \cos \theta + \cos 45^\circ \sin \theta \]

### Step 2: Substitute Values for Sine and Cosine
We know that:

\[ \sin 45^\circ = \cos 45^\circ = \frac{1}{\sqrt{2}} \]

Thus, substituting these values into the equation gives:

\[ \sin(45^\circ + \theta) = \frac{1}{\sqrt{2}} \cos \theta + \frac{1}{\sqrt{2}} \sin \theta \]

### Step 3: Apply the Cosine Subtraction Formula
Now, we apply the cosine subtraction formula:

\[ \cos(A - B) = \cos A \cos B + \sin A \sin B \]

For \(A = 45^\circ\) and \(B = \theta\):

\[ \cos(45^\circ - \theta) = \cos 45^\circ \cos \theta + \sin 45^\circ \sin \theta \]

### Step 4: Substitute Values for Sine and Cosine Again
Again substituting the known values:

\[ \cos(45^\circ - \theta) = \frac{1}{\sqrt{2}} \cos \theta + \frac{1}{\sqrt{2}} \sin \theta \]

### Step 5: Substitute Back into the Original Expression
Now we substitute both results back into the original expression:

\[ \sin(45^\circ + \theta) - \cos(45^\circ - \theta) = \left(\frac{1}{\sqrt{2}} \cos \theta + \frac{1}{\sqrt{2}} \sin \theta\right) - \left(\frac{1}{\sqrt{2}} \cos \theta + \frac{1}{\sqrt{2}} \sin \theta\right) \]

### Step 6: Simplify the Expression
When we simplify this, we see that:

\[ \sin(45^\circ + \theta) - \cos(45^\circ - \theta) = 0 \]

### Final Answer
Thus, the value of the expression is:

\[ \boxed{0} \]

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