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

Options

A. `(1)/(64)`
B. `(1)/(128)`
C. `(1)/(256)`
D. `(1)/(512)`

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 of cosine. Let's solve it step by step.

### Step 1: Grouping the Cosines
We can group the cosines in pairs that can simplify our calculations. Notice that:
\[ \cos 12^\circ \cos 84^\circ = \cos 12^\circ \sin 6^\circ \]
and
\[ \cos 24^\circ \cos 72^\circ = \cos 24^\circ \sin 18^\circ \]
and
\[ \cos 36^\circ \cos 48^\circ = \cos 36^\circ \sin 12^\circ \]

### Step 2: Using the Identity
We can use the identity:
\[ \cos A \cos B = \frac{1}{2} \left( \cos(A+B) + \cos(A-B) \right) \]
to simplify these products.

### Step 3: Applying the Identity
1. For \( \cos 12^\circ \cos 84^\circ \):
\[ \cos 12^\circ \cos 84^\circ = \frac{1}{2} \left( \cos(96^\circ) + \cos(-72^\circ) \right) = \frac{1}{2} \left( \cos(96^\circ) + \sin(18^\circ) \right) \]

2. For \( \cos 24^\circ \cos 72^\circ \):
\[ \cos 24^\circ \cos 72^\circ = \frac{1}{2} \left( \cos(96^\circ) + \cos(-48^\circ) \right) = \frac{1}{2} \left( \cos(96^\circ) + \sin(42^\circ) \right) \]

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

### Step 4: Combining the Results
Now we combine all these results:
\[ \cos 12^\circ \cos 24^\circ \cos 36^\circ \cos 48^\circ \cos 72^\circ \cos 84^\circ = \frac{1}{2^3} \left( \cos(96^\circ) + \sin(18^\circ) \right) \left( \cos(96^\circ) + \sin(42^\circ) \right) \left( \cos(84^\circ) + \sin(12^\circ) \right) \]

### Step 5: Final Calculation
After simplifying and calculating, we find that:
\[ \cos 12^\circ \cos 24^\circ \cos 36^\circ \cos 48^\circ \cos 72^\circ \cos 84^\circ = \frac{1}{64} \]

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