find
`cos175^(@)+cos65^(@)+cos55^(@)`

Options

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

Explanation

To solve the problem of finding \( \cos 175^\circ + \cos 65^\circ + \cos 55^\circ \), we can use the cosine addition formula and properties of cosine. Let's go through the solution step by step.

### Step 1: Group the terms
We start with the expression:
\[ \cos 175^\circ + \cos 65^\circ + \cos 55^\circ \]

### Step 2: Use the cosine addition formula
We can group the first two terms:
\[ \cos 175^\circ + \cos 65^\circ \]
Using the formula for the sum of cosines:
\[ \cos A + \cos B = 2 \cos\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \]
Let \( A = 175^\circ \) and \( B = 65^\circ \):
\[ \cos 175^\circ + \cos 65^\circ = 2 \cos\left(\frac{175 + 65}{2}\right) \cos\left(\frac{175 - 65}{2}\right) \]

### Step 3: Calculate the averages
Calculating the averages:
\[ \frac{175 + 65}{2} = \frac{240}{2} = 120^\circ \]
\[ \frac{175 - 65}{2} = \frac{110}{2} = 55^\circ \]
Thus, we have:
\[ \cos 175^\circ + \cos 65^\circ = 2 \cos(120^\circ) \cos(55^\circ) \]

### Step 4: Substitute the value of \( \cos(120^\circ) \)
We know that:
\[ \cos(120^\circ) = -\frac{1}{2} \]
So substituting this value:
\[ \cos 175^\circ + \cos 65^\circ = 2 \left(-\frac{1}{2}\right) \cos(55^\circ) = -\cos(55^\circ) \]

### Step 5: Combine with the remaining term
Now we add \( \cos 55^\circ \):
\[ -\cos(55^\circ) + \cos(55^\circ) = 0 \]

### Conclusion
Thus, we have shown that:
\[ \cos 175^\circ + \cos 65^\circ + \cos 55^\circ = 0 \]