The value of `cos 75^(@) cos 15^(@)` is equal to

Options

A. `1/2`
B. `(1)/(sqrt2)`
C. `1/4`
D. 1

Answer

To find the value of \( \cos 75^\circ \cos 15^\circ \), we can use the product-to-sum identities. Here’s a step-by-step solution:

### Step 1: Use the product-to-sum identity
The product-to-sum identity states that:
\[ 2 \cos A \cos B = \cos(A - B) + \cos(A + B) \]
In our case, we have \( A = 75^\circ \) and \( B = 15^\circ \).

### Step 2: Rewrite the expression
To apply the identity, we can multiply and divide by 2:
\[ \cos 75^\circ \cos 15^\circ = \frac{1}{2} \cdot 2 \cos 75^\circ \cos 15^\circ = \frac{1}{2} \left( \cos(75^\circ - 15^\circ) + \cos(75^\circ + 15^\circ) \right) \]

### Step 3: Calculate the angles
Now, calculate \( 75^\circ - 15^\circ \) and \( 75^\circ + 15^\circ \):
\[ 75^\circ - 15^\circ = 60^\circ \]
\[ 75^\circ + 15^\circ = 90^\circ \]

### Step 4: Substitute back into the expression
Now substitute these values back into the equation:
\[ \cos 75^\circ \cos 15^\circ = \frac{1}{2} \left( \cos 60^\circ + \cos 90^\circ \right) \]

### Step 5: Use known values of cosine
We know that:
\[ \cos 60^\circ = \frac{1}{2} \quad \text{and} \quad \cos 90^\circ = 0 \]
Substituting these values gives:
\[ \cos 75^\circ \cos 15^\circ = \frac{1}{2} \left( \frac{1}{2} + 0 \right) = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4} \]

### Final Answer
Thus, the value of \( \cos 75^\circ \cos 15^\circ \) is:
\[ \frac{1}{4} \]