The value of `cos15^(@)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, we need to find the value of \( \cos(15^\circ) \cos\left(7\frac{1}{2}^\circ\right) \sin\left(7\frac{1}{2}^\circ\right) \).

### Step 1: Rewrite the expression
We can use the identity for \( \sin(2\theta) \):
\[ \sin(2\theta) = 2 \sin(\theta) \cos(\theta) \]
In our case, we can let \( \theta = 7.5^\circ \). Therefore:
\[ \sin(15^\circ) = 2 \sin(7.5^\circ) \cos(7.5^\circ) \]
This means:
\[ \sin(7.5^\circ) \cos(7.5^\circ) = \frac{1}{2} \sin(15^\circ) \]

### Step 2: Substitute back into the original expression
Now we can substitute this back into our original expression:
\[ \cos(15^\circ) \cos(7.5^\circ) \sin(7.5^\circ) = \cos(15^\circ) \cdot \frac{1}{2} \sin(15^\circ) \]

### Step 3: Simplify the expression
This simplifies to:
\[ \frac{1}{2} \cos(15^\circ) \sin(15^\circ) \]

### Step 4: Use the identity for \( \sin(2\theta) \) again
We can use the identity again:
\[ \sin(2\theta) = 2 \sin(\theta) \cos(\theta) \]
Thus:
\[ \sin(30^\circ) = 2 \sin(15^\circ) \cos(15^\circ) \]
We know that \( \sin(30^\circ) = \frac{1}{2} \), so:
\[ \frac{1}{2} \sin(30^\circ) = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4} \]

### Final Result
Thus, the value of \( \cos(15^\circ) \cos\left(7\frac{1}{2}^\circ\right) \sin\left(7\frac{1}{2}^\circ\right) \) is:
\[ \frac{1}{4} \]