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

Options

A. `(1)/(sqrt2)`
B. `(1)/(2)`
C. `-(1)/(sqrt2)`
D. 0

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: Use the co-function identity
Recall the co-function identity for sine:
\[ \sin \theta = \cos(90^\circ - \theta) \]
Using this identity, we can rewrite \( \sin 15^\circ \):
\[ \sin 15^\circ = \cos(90^\circ - 15^\circ) = \cos 75^\circ \]
Thus, we can rewrite our expression:
\[ \cos 15^\circ - \sin 15^\circ = \cos 15^\circ - \cos 75^\circ \]

### Step 3: Use the cosine subtraction formula
We can use the cosine subtraction formula:
\[ \cos A - \cos B = -2 \sin\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right) \]
Let \( A = 15^\circ \) and \( B = 75^\circ \):
\[ \cos 15^\circ - \cos 75^\circ = -2 \sin\left(\frac{15^\circ + 75^\circ}{2}\right) \sin\left(\frac{15^\circ - 75^\circ}{2}\right) \]

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

### Step 5: Substitute back into the formula
Now substituting back into the formula:
\[ \cos 15^\circ - \cos 75^\circ = -2 \sin(45^\circ) \sin(-30^\circ) \]

### Step 6: Use known values of sine
We know:
\[ \sin(45^\circ) = \frac{\sqrt{2}}{2}, \quad \sin(-30^\circ) = -\frac{1}{2} \]
Substituting these values:
\[ \cos 15^\circ - \cos 75^\circ = -2 \left(\frac{\sqrt{2}}{2}\right) \left(-\frac{1}{2}\right) \]
\[ = -2 \cdot \frac{\sqrt{2}}{2} \cdot -\frac{1}{2} = \frac{\sqrt{2}}{2} \]

### Step 7: Final value
Thus, the value of \( \cos 15^\circ - \sin 15^\circ \) is:
\[ \frac{\sqrt{2}}{2} \]