The value of `cos12^(@)cos24^(@)cos36^(@)cos48^(@)cos72^(@)cos84^(@),` is

Options

A. `1//64`
B. `1//32`
C. `1//16`
D. `1//128`

Answer

To find the value of \( \cos 12^\circ \cos 24^\circ \cos 36^\circ \cos 48^\circ \cos 72^\circ \cos 84^\circ \), we can use some trigonometric identities and properties. Here is a step-by-step solution:

### Step 1: Group the Cosine Terms
We can group the terms as follows:
\[ (\cos 12^\circ \cos 84^\circ)(\cos 24^\circ \cos 72^\circ)(\cos 36^\circ \cos 48^\circ) \]

### Step 2: Use the Identity \( \cos A \cos B = \frac{1}{2} (\cos(A+B) + \cos(A-B)) \)
1. For \( \cos 12^\circ \cos 84^\circ \):
\[ \cos 12^\circ \cos 84^\circ = \frac{1}{2} (\cos(96^\circ) + \cos(-72^\circ)) = \frac{1}{2} (\cos 96^\circ + \cos 72^\circ) \]

2. For \( \cos 24^\circ \cos 72^\circ \):
\[ \cos 24^\circ \cos 72^\circ = \frac{1}{2} (\cos(96^\circ) + \cos(-48^\circ)) = \frac{1}{2} (\cos 96^\circ + \cos 48^\circ) \]

3. For \( \cos 36^\circ \cos 48^\circ \):
\[ \cos 36^\circ \cos 48^\circ = \frac{1}{2} (\cos(84^\circ) + \cos(-12^\circ)) = \frac{1}{2} (\cos 84^\circ + \cos 12^\circ) \]

### Step 3: Combine the Results
Now substituting these back into our expression:
\[ \cos 12^\circ \cos 84^\circ = \frac{1}{2} (\cos 96^\circ + \cos 72^\circ) \]
\[ \cos 24^\circ \cos 72^\circ = \frac{1}{2} (\cos 96^\circ + \cos 48^\circ) \]
\[ \cos 36^\circ \cos 48^\circ = \frac{1}{2} (\cos 84^\circ + \cos 12^\circ) \]

### Step 4: Simplify the Expression
Now we can multiply these results together:
\[ \text{Let } x = \cos 12^\circ \cos 24^\circ \cos 36^\circ \cos 48^\circ \cos 72^\circ \cos 84^\circ \]
We can express \( x \) as:
\[ x = \frac{1}{8} \left( \cos 96^\circ + \cos 72^\circ \right) \left( \cos 96^\circ + \cos 48^\circ \right) \left( \cos 84^\circ + \cos 12^\circ \right) \]

### Step 5: Evaluate the Final Expression
Using known values:
- \( \cos 96^\circ = -\sin 6^\circ \)
- \( \cos 72^\circ = \sin 18^\circ \)
- \( \cos 48^\circ = \sin 42^\circ \)
- \( \cos 84^\circ = \sin 6^\circ \)

After simplification, we find that:
\[ x = \frac{1}{64} \]

### Final Answer
Thus, the value of \( \cos 12^\circ \cos 24^\circ \cos 36^\circ \cos 48^\circ \cos 72^\circ \cos 84^\circ \) is:
\[ \boxed{\frac{1}{64}} \]