The value of ` cos 15^(@) cos 7 ((1)/(2))^(@)sin 7 ((1)/(2))^(@)` is

Options

A. `(1)/(2)`
B. `(1)/(8)`
C. `(1)/(4)`
D. `(1)/(16)`

Answer

To solve the problem of finding the value of \( \cos 15^\circ \cos 7.5^\circ \sin 7.5^\circ \), we will follow these steps:

### Step 1: Write the expression
We start with the expression:
\[ \cos 15^\circ \cos 7.5^\circ \sin 7.5^\circ \]

### Step 2: Use the identity for \( \sin 2\theta \)
We can use the identity:
\[ \sin 2\theta = 2 \sin \theta \cos \theta \]
In our case, we can set \( \theta = 7.5^\circ \):
\[ \sin 15^\circ = 2 \sin 7.5^\circ \cos 7.5^\circ \]
Thus, we can rewrite our expression as:
\[ \cos 15^\circ \cdot \frac{1}{2} \sin 15^\circ \]

### Step 3: Rewrite the expression
Now, we can rewrite our expression:
\[ \cos 15^\circ \cdot \frac{1}{2} \sin 15^\circ = \frac{1}{2} \cos 15^\circ \sin 15^\circ \]

### Step 4: Use the identity for \( \sin 2\theta \) again
Now we can apply the identity again, but this time for \( \sin 30^\circ \):
\[ \sin 30^\circ = 2 \sin 15^\circ \cos 15^\circ \]
Thus, we have:
\[ \frac{1}{2} \cos 15^\circ \sin 15^\circ = \frac{1}{4} \sin 30^\circ \]

### Step 5: Calculate \( \sin 30^\circ \)
We know that:
\[ \sin 30^\circ = \frac{1}{2} \]
So we can substitute this value back into our expression:
\[ \frac{1}{4} \sin 30^\circ = \frac{1}{4} \cdot \frac{1}{2} = \frac{1}{8} \]

### Final Answer
Thus, the value of \( \cos 15^\circ \cos 7.5^\circ \sin 7.5^\circ \) is:
\[ \frac{1}{8} \]