The value of `cos 65^(@)cos 55^(@)cos5^(@)` is

Options

A. `(sqrt(3)+1)/(8sqrt(2))`
B. `(sqrt(3)-1)/(8sqrt(2))`
C. `(sqrt(3)+1)/(4sqrt(2))`
D. `(sqrt(3)-1)/(4sqrt(2))`

Explanation

To find the value of \( \cos 65^\circ \cos 55^\circ \cos 5^\circ \), we can use trigonometric identities to simplify the expression. Here’s a step-by-step solution:

### Step 1: Multiply by 2 and divide by 2
We start by rewriting the expression:
\[ \cos 65^\circ \cos 55^\circ \cos 5^\circ = \frac{1}{2} \cdot 2 \cos 65^\circ \cos 55^\circ \cos 5^\circ \]

### Step 2: Use the product-to-sum identities
We can use the product-to-sum identities to simplify \( 2 \cos A \cos B \):
\[ 2 \cos A \cos B = \cos(A + B) + \cos(A - B) \]
Let \( A = 65^\circ \) and \( B = 55^\circ \):
\[ 2 \cos 65^\circ \cos 55^\circ = \cos(65^\circ + 55^\circ) + \cos(65^\circ - 55^\circ) \]
Calculating the angles:
\[ 65^\circ + 55^\circ = 120^\circ \quad \text{and} \quad 65^\circ - 55^\circ = 10^\circ \]
Thus, we have:
\[ 2 \cos 65^\circ \cos 55^\circ = \cos 120^\circ + \cos 10^\circ \]

### Step 3: Substitute back into the equation
Now substituting back, we get:
\[ \cos 65^\circ \cos 55^\circ \cos 5^\circ = \frac{1}{2} \left( \cos 120^\circ + \cos 10^\circ \right) \cos 5^\circ \]

### Step 4: Calculate \( \cos 120^\circ \)
We know that:
\[ \cos 120^\circ = -\frac{1}{2} \]
So, substituting this value in:
\[ \cos 65^\circ \cos 55^\circ \cos 5^\circ = \frac{1}{2} \left( -\frac{1}{2} + \cos 10^\circ \right) \cos 5^\circ \]

### Step 5: Simplify further
This simplifies to:
\[ \cos 65^\circ \cos 55^\circ \cos 5^\circ = \frac{1}{2} \left( -\frac{1}{2} \cos 5^\circ + \cos 10^\circ \cos 5^\circ \right) \]

### Step 6: Use product-to-sum again
Now, we apply the product-to-sum identity again on \( \cos 10^\circ \cos 5^\circ \):
\[ 2 \cos 10^\circ \cos 5^\circ = \cos(10^\circ + 5^\circ) + \cos(10^\circ - 5^\circ) = \cos 15^\circ + \cos 5^\circ \]
Thus:
\[ \cos 10^\circ \cos 5^\circ = \frac{1}{2} (\cos 15^\circ + \cos 5^\circ) \]

### Step 7: Substitute back
Substituting this back gives:
\[ \cos 65^\circ \cos 55^\circ \cos 5^\circ = \frac{1}{2} \left( -\frac{1}{2} \cos 5^\circ + \frac{1}{2} (\cos 15^\circ + \cos 5^\circ) \right) \]
This simplifies to:
\[ \cos 65^\circ \cos 55^\circ \cos 5^\circ = \frac{1}{4} \cos 15^\circ \]

### Final Result
Thus, the value of \( \cos 65^\circ \cos 55^\circ \cos 5^\circ \) is:
\[ \frac{1}{4} \cos 15^\circ \]