The value of `cos15^(@)-sin15^(@)` is

Options

A. `(1)/(sqrt(2))`
B. `(1)/(2)`
C. `-(1)/(sqrt(2))`
D. 0

Answer

To find the value of \( \cos 15^\circ - \sin 15^\circ \), we can follow these steps:

### Step 1: Multiply and Divide by \( \sqrt{2} \)
We start by rewriting the expression:
\[ \cos 15^\circ - \sin 15^\circ = \frac{\sqrt{2}}{\sqrt{2}} (\cos 15^\circ - \sin 15^\circ) = \frac{\sqrt{2} (\cos 15^\circ - \sin 15^\circ)}{\sqrt{2}} \]

### Step 2: Use the Angle Sum and Difference Identity
We can express \( \cos 15^\circ - \sin 15^\circ \) in terms of sine:
\[ \sqrt{2} (\cos 15^\circ - \sin 15^\circ) = \sqrt{2} \cos 15^\circ - \sqrt{2} \sin 15^\circ \]
This can be rewritten using the sine of a difference:
\[ \sqrt{2} (\cos 15^\circ - \sin 15^\circ) = \sqrt{2} \left( \cos 15^\circ \cdot \frac{1}{\sqrt{2}} - \sin 15^\circ \cdot \frac{1}{\sqrt{2}} \right) \]
Recognizing that \( \frac{1}{\sqrt{2}} = \sin 45^\circ = \cos 45^\circ \), we can express this as:
\[ \sqrt{2} (\cos 15^\circ - \sin 15^\circ) = \sqrt{2} \left( \sin 45^\circ \cos 15^\circ - \cos 45^\circ \sin 15^\circ \right) \]
Using the sine subtraction formula, we have:
\[ \sqrt{2} (\cos 15^\circ - \sin 15^\circ) = \sqrt{2} \sin(45^\circ - 15^\circ) = \sqrt{2} \sin 30^\circ \]

### Step 3: Calculate \( \sin 30^\circ \)
We know that:
\[ \sin 30^\circ = \frac{1}{2} \]
Thus, substituting this back, we get:
\[ \sqrt{2} \sin 30^\circ = \sqrt{2} \cdot \frac{1}{2} = \frac{\sqrt{2}}{2} \]

### Step 4: Final Expression
Now, we substitute back to find:
\[ \cos 15^\circ - \sin 15^\circ = \frac{\sqrt{2}}{2} \]

### Conclusion
Thus, the value of \( \cos 15^\circ - \sin 15^\circ \) is:
\[ \cos 15^\circ - \sin 15^\circ = \frac{1}{\sqrt{2}} \]