What is the value of `cos306^(@)+cos234^(@)+cos162^(@)+cos18^(@)`?

Options

A. 1
B. `-1`
C. 0
D. 2

Explanation

To find the value of \( \cos 306^\circ + \cos 234^\circ + \cos 162^\circ + \cos 18^\circ \), we can simplify each term using trigonometric identities. Let's break it down step by step.

### Step 1: Rewrite the angles
We can express the angles in terms of their reference angles:
- \( \cos 306^\circ = \cos(360^\circ - 54^\circ) = \cos 54^\circ \)
- \( \cos 234^\circ = \cos(180^\circ + 54^\circ) = -\cos 54^\circ \)
- \( \cos 162^\circ = \cos(180^\circ - 18^\circ) = -\cos 18^\circ \)
- \( \cos 18^\circ = \cos 18^\circ \)

### Step 2: Substitute the rewritten angles
Now we can substitute these values back into the expression:
\[ \cos 306^\circ + \cos 234^\circ + \cos 162^\circ + \cos 18^\circ = \cos 54^\circ - \cos 54^\circ - \cos 18^\circ + \cos 18^\circ \]

### Step 3: Simplify the expression
Now, let's simplify the expression:
\[ \cos 54^\circ - \cos 54^\circ - \cos 18^\circ + \cos 18^\circ = 0 \]

### Conclusion
Thus, the value of \( \cos 306^\circ + \cos 234^\circ + \cos 162^\circ + \cos 18^\circ \) is:
\[ \boxed{0} \]