The value of `cos12^@+cos84^@+cos156^@+cos132^@` is

Options

A. `(1)/(2)`
B. 1
C. `-(1)/(2)`
D. `(1)/(8)`

Explanation

To find the value of \( \cos 12^\circ + \cos 84^\circ + \cos 156^\circ + \cos 132^\circ \), we can follow these steps:

### Step 1: Group the terms
We can rearrange the terms as follows:
\[ \cos 12^\circ + \cos 132^\circ + \cos 84^\circ + \cos 156^\circ \]

### Step 2: Apply the cosine addition formula
We use the formula:
\[ \cos C + \cos D = 2 \cos\left(\frac{C + D}{2}\right) \cos\left(\frac{C - D}{2}\right) \]
First, we group \( \cos 12^\circ \) and \( \cos 132^\circ \):
\[ \cos 12^\circ + \cos 132^\circ = 2 \cos\left(\frac{12^\circ + 132^\circ}{2}\right) \cos\left(\frac{12^\circ - 132^\circ}{2}\right) \]
Calculating the angles:
\[ \frac{12^\circ + 132^\circ}{2} = \frac{144^\circ}{2} = 72^\circ \]
\[ \frac{12^\circ - 132^\circ}{2} = \frac{-120^\circ}{2} = -60^\circ \]
Thus,
\[ \cos 12^\circ + \cos 132^\circ = 2 \cos 72^\circ \cos(-60^\circ) \]

### Step 3: Simplify using cosine properties
Using \( \cos(-\theta) = \cos(\theta) \):
\[ \cos(-60^\circ) = \cos(60^\circ) = \frac{1}{2} \]
So,
\[ \cos 12^\circ + \cos 132^\circ = 2 \cos 72^\circ \cdot \frac{1}{2} = \cos 72^\circ \]

### Step 4: Now group the other two terms
Next, we group \( \cos 84^\circ \) and \( \cos 156^\circ \):
\[ \cos 84^\circ + \cos 156^\circ = 2 \cos\left(\frac{84^\circ + 156^\circ}{2}\right) \cos\left(\frac{84^\circ - 156^\circ}{2}\right) \]
Calculating the angles:
\[ \frac{84^\circ + 156^\circ}{2} = \frac{240^\circ}{2} = 120^\circ \]
\[ \frac{84^\circ - 156^\circ}{2} = \frac{-72^\circ}{2} = -36^\circ \]
Thus,
\[ \cos 84^\circ + \cos 156^\circ = 2 \cos 120^\circ \cos(-36^\circ) \]

### Step 5: Simplify using cosine properties
Using \( \cos(-\theta) = \cos(\theta) \):
\[ \cos(-36^\circ) = \cos(36^\circ) \]
And knowing \( \cos 120^\circ = -\frac{1}{2} \):
\[ \cos 84^\circ + \cos 156^\circ = 2 \left(-\frac{1}{2}\right) \cos 36^\circ = -\cos 36^\circ \]

### Step 6: Combine results
Now we combine the results:
\[ \cos 12^\circ + \cos 84^\circ + \cos 156^\circ + \cos 132^\circ = \cos 72^\circ - \cos 36^\circ \]

### Step 7: Substitute known values
We know:
\[ \cos 72^\circ = \sin 18^\circ \quad \text{and} \quad \cos 36^\circ = \sin 54^\circ \]
Using \( \sin 54^\circ = \cos 36^\circ \):
\[ \cos 72^\circ - \cos 36^\circ = \sin 18^\circ - \sin 54^\circ \]

### Step 8: Calculate the final value
Using the known values:
\[ \sin 18^\circ = \frac{\sqrt{5}-1}{4} \quad \text{and} \quad \cos 36^\circ = \frac{\sqrt{5}+1}{4} \]
Thus,
\[ \sin 18^\circ - \cos 36^\circ = \frac{\sqrt{5}-1}{4} - \frac{\sqrt{5}+1}{4} = \frac{\sqrt{5}-1 - \sqrt{5} - 1}{4} = \frac{-2}{4} = -\frac{1}{2} \]

### Final Answer
The value of \( \cos 12^\circ + \cos 84^\circ + \cos 156^\circ + \cos 132^\circ \) is:
\[ \boxed{-\frac{1}{2}} \]