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 and formulas. Here’s a step-by-step solution:

### Step 1: Rewrite the expression
We start with the expression:
\[ \cos 65^\circ \cos 55^\circ \cos 5^\circ \]

### Step 2: Use the product-to-sum formula
We can apply the product-to-sum formula for two cosines:
\[ \cos A \cos B = \frac{1}{2} \left( \cos(A + B) + \cos(A - B) \right) \]
Let's first combine \( \cos 55^\circ \) and \( \cos 5^\circ \):
\[ \cos 55^\circ \cos 5^\circ = \frac{1}{2} \left( \cos(55^\circ + 5^\circ) + \cos(55^\circ - 5^\circ) \right) \]
This simplifies to:
\[ \cos 55^\circ \cos 5^\circ = \frac{1}{2} \left( \cos 60^\circ + \cos 50^\circ \right) \]

### Step 3: Substitute back into the expression
Now substituting this back into the original expression:
\[ \cos 65^\circ \cos 55^\circ \cos 5^\circ = \cos 65^\circ \cdot \frac{1}{2} \left( \cos 60^\circ + \cos 50^\circ \right) \]
This can be rewritten as:
\[ = \frac{1}{2} \cos 65^\circ \cos 60^\circ + \frac{1}{2} \cos 65^\circ \cos 50^\circ \]

### Step 4: Evaluate \( \cos 60^\circ \)
We know that:
\[ \cos 60^\circ = \frac{1}{2} \]
Thus:
\[ \frac{1}{2} \cos 65^\circ \cos 60^\circ = \frac{1}{2} \cdot \frac{1}{2} \cos 65^\circ = \frac{1}{4} \cos 65^\circ \]

### Step 5: Evaluate \( \cos 65^\circ \cos 50^\circ \)
Now, we can apply the product-to-sum formula again on \( \cos 65^\circ \cos 50^\circ \):
\[ \cos 65^\circ \cos 50^\circ = \frac{1}{2} \left( \cos(65^\circ + 50^\circ) + \cos(65^\circ - 50^\circ) \right) \]
This simplifies to:
\[ \cos 65^\circ \cos 50^\circ = \frac{1}{2} \left( \cos 115^\circ + \cos 15^\circ \right) \]

### Step 6: Substitute back
Substituting this back gives:
\[ \frac{1}{2} \cdot \frac{1}{2} \left( \cos 115^\circ + \cos 15^\circ \right) = \frac{1}{4} \left( \cos 115^\circ + \cos 15^\circ \right) \]

### Step 7: Combine the terms
Now, combining all the terms:
\[ \cos 65^\circ \cos 55^\circ \cos 5^\circ = \frac{1}{4} \cos 65^\circ + \frac{1}{4} \left( \cos 115^\circ + \cos 15^\circ \right) \]

### Step 8: Evaluate \( \cos 115^\circ \)
Using the fact that \( \cos 115^\circ = -\cos 65^\circ \):
\[ \cos 65^\circ \cos 55^\circ \cos 5^\circ = \frac{1}{4} \cos 65^\circ + \frac{1}{4} \left( -\cos 65^\circ + \cos 15^\circ \right) \]
This simplifies to:
\[ = \frac{1}{4} \cos 15^\circ \]

### Step 9: Final Result
Thus, the final value is:
\[ \cos 65^\circ \cos 55^\circ \cos 5^\circ = \frac{1}{4} \cos 15^\circ \]