`cos175^(@)+cos65^(@)+cos55^(@)=?`

Options

A. 2
B. 0
C. 1
D. None of these

Answer

To solve the expression \( \cos 175^\circ + \cos 65^\circ + \cos 55^\circ \), we will use the cosine addition formula.

### Step-by-Step Solution:

1. **Write the given expression:**
\[ \cos 175^\circ + \cos 65^\circ + \cos 55^\circ \]

2. **Group the first two terms:**
We can use the formula for the sum of cosines:
\[ \cos A + \cos B = 2 \cos\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right) \]
Here, let \( A = 175^\circ \) and \( B = 65^\circ \).

3. **Calculate \( A + B \) and \( A - B \):**
\[ A + B = 175^\circ + 65^\circ = 240^\circ \]
\[ A - B = 175^\circ - 65^\circ = 110^\circ \]

4. **Substitute into the formula:**
\[ \cos 175^\circ + \cos 65^\circ = 2 \cos\left(\frac{240^\circ}{2}\right) \cos\left(\frac{110^\circ}{2}\right) \]
\[ = 2 \cos(120^\circ) \cos(55^\circ) \]

5. **Evaluate \( \cos(120^\circ) \):**
\[ \cos(120^\circ) = -\frac{1}{2} \]
Therefore,
\[ 2 \cos(120^\circ) \cos(55^\circ) = 2 \left(-\frac{1}{2}\right) \cos(55^\circ) = -\cos(55^\circ) \]

6. **Combine with the third term:**
Now, we add \( \cos 55^\circ \):
\[ -\cos(55^\circ) + \cos(55^\circ) = 0 \]

### Final Answer:
\[ \cos 175^\circ + \cos 65^\circ + \cos 55^\circ = 0 \]