The value of ` cos 15^(@) - sin 15^(@)` is

Options

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

Answer

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

### Step 1: Write the expression
We start with the expression:
\[ \cos 15^\circ - \sin 15^\circ \]

### Step 2: Rewrite \( \cos 15^\circ \)
We can express \( \cos 15^\circ \) using the identity \( \cos(90^\circ - x) = \sin x \):
\[ \cos 15^\circ = \sin(90^\circ - 75^\circ) = \sin 75^\circ \]
Thus, we can rewrite the expression as:
\[ \sin 75^\circ - \sin 15^\circ \]

### Step 3: Apply the sine difference formula
We can use the sine difference identity:
\[ \sin A - \sin B = 2 \cos\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right) \]
Here, let \( A = 75^\circ \) and \( B = 15^\circ \):
\[ \sin 75^\circ - \sin 15^\circ = 2 \cos\left(\frac{75^\circ + 15^\circ}{2}\right) \sin\left(\frac{75^\circ - 15^\circ}{2}\right) \]

### Step 4: Calculate the averages
Calculating the averages:
\[ \frac{75^\circ + 15^\circ}{2} = \frac{90^\circ}{2} = 45^\circ \]
\[ \frac{75^\circ - 15^\circ}{2} = \frac{60^\circ}{2} = 30^\circ \]

### Step 5: Substitute back into the equation
Substituting these values back into the equation:
\[ \sin 75^\circ - \sin 15^\circ = 2 \cos(45^\circ) \sin(30^\circ) \]

### Step 6: Use known values
Now, we use the known values:
\[ \cos(45^\circ) = \frac{1}{\sqrt{2}}, \quad \sin(30^\circ) = \frac{1}{2} \]
Thus:
\[ \sin 75^\circ - \sin 15^\circ = 2 \cdot \frac{1}{\sqrt{2}} \cdot \frac{1}{2} \]

### Step 7: Simplify the expression
Simplifying gives:
\[ = \frac{2}{2\sqrt{2}} = \frac{1}{\sqrt{2}} \]

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