`cos24^(@)+cos55^(@)+cos125^(@)+cos204^(@)+cos300^(@)=?`

Options

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

Answer

To solve the equation \( \cos 24^\circ + \cos 55^\circ + \cos 125^\circ + \cos 204^\circ + \cos 300^\circ \), we can use properties of cosine and symmetry in the unit circle.

### Step-by-Step Solution:

1. **Identify Cosine Relationships**:
- We know that \( \cos(180^\circ - x) = -\cos x \) and \( \cos(180^\circ + x) = -\cos x \).
- We can rewrite \( \cos 125^\circ \) and \( \cos 204^\circ \) using these properties.

2. **Rewrite \( \cos 125^\circ \)**:
- \( \cos 125^\circ = \cos(180^\circ - 55^\circ) = -\cos 55^\circ \).

3. **Rewrite \( \cos 204^\circ \)**:
- \( \cos 204^\circ = \cos(180^\circ + 24^\circ) = -\cos 24^\circ \).

4. **Substitute Back into the Equation**:
- Now substitute these values back into the original equation:
\[ \cos 24^\circ + \cos 55^\circ + (-\cos 55^\circ) + (-\cos 24^\circ) + \cos 300^\circ \]

5. **Simplify the Equation**:
- The equation simplifies to:
\[ \cos 24^\circ - \cos 24^\circ + \cos 55^\circ - \cos 55^\circ + \cos 300^\circ \]
- This reduces to:
\[ 0 + \cos 300^\circ \]

6. **Calculate \( \cos 300^\circ \)**:
- \( \cos 300^\circ = \cos(360^\circ - 60^\circ) = \cos 60^\circ = \frac{1}{2} \).

7. **Final Result**:
- Therefore, the final answer is:
\[ \cos 24^\circ + \cos 55^\circ + \cos 125^\circ + \cos 204^\circ + \cos 300^\circ = \frac{1}{2} \]