`cos15^(@)cos7""(1)/(2)""^(@).cos82""(1)/(2)""^(@)=?`

Options

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

Answer

To solve the problem \( \cos 15^\circ \cos 7.5^\circ \cos 82.5^\circ \), we can follow these steps:

### Step 1: Rewrite the angles
First, let's rewrite the angles in a simpler form:
\[ \cos 15^\circ \cos 7.5^\circ \cos 82.5^\circ \]

### Step 2: Use the identity for cosine
We can use the identity \( \cos \theta = \sin(90^\circ - \theta) \) to rewrite \( \cos 82.5^\circ \):
\[ \cos 82.5^\circ = \sin(90^\circ - 82.5^\circ) = \sin 7.5^\circ \]
Now we can rewrite the expression:
\[ \cos 15^\circ \cos 7.5^\circ \sin 7.5^\circ \]

### Step 3: Apply the double angle identity
Next, we can apply the double angle identity. We know that:
\[ 2 \sin a \cos a = \sin(2a) \]
So, we can write:
\[ \sin 7.5^\circ \cos 7.5^\circ = \frac{1}{2} \sin(15^\circ) \]
Thus, our expression becomes:
\[ \cos 15^\circ \cdot \frac{1}{2} \sin(15^\circ) \]

### Step 4: Simplify the expression
Now we can simplify:
\[ \frac{1}{2} \cos 15^\circ \sin 15^\circ \]
Again, we use the double angle identity:
\[ \sin(2a) = 2 \sin a \cos a \]
So:
\[ \sin(30^\circ) = 2 \sin(15^\circ) \cos(15^\circ) \]
This means:
\[ \frac{1}{2} \sin(30^\circ) \]

### Step 5: Calculate the final value
We know that:
\[ \sin(30^\circ) = \frac{1}{2} \]
Thus:
\[ \frac{1}{2} \sin(30^\circ) = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4} \]

### Step 6: Final answer
The final answer is:
\[ \frac{1}{4} \]