Which one of the following is the value of cos `170^@ cos 10^@ - sin 170^@ sin 10^@`

Options

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

Answer

To solve the expression \( \cos 170^\circ \cos 10^\circ - \sin 170^\circ \sin 10^\circ \), we can use the cosine addition formula:

### Step-by-Step Solution:

1. **Identify the formula**:
The cosine addition formula states that:
\[ \cos(A + B) = \cos A \cos B - \sin A \sin B \]
Here, we can identify \( A = 170^\circ \) and \( B = 10^\circ \).

2. **Apply the formula**:
By substituting \( A \) and \( B \) into the formula, we have:
\[ \cos(170^\circ + 10^\circ) = \cos 170^\circ \cos 10^\circ - \sin 170^\circ \sin 10^\circ \]
This simplifies to:
\[ \cos 180^\circ \]

3. **Evaluate \( \cos 180^\circ \)**:
We know that:
\[ \cos 180^\circ = -1 \]

4. **Conclusion**:
Therefore, the value of \( \cos 170^\circ \cos 10^\circ - \sin 170^\circ \sin 10^\circ \) is:
\[ -1 \]

### Final Answer:
The value is \(-1\).