`cos 38^(@) cos 8^(@) + sin 38^(@)sin 8^(@) ` is equal to

Options

A. `cos 30^(@)`
B. `cos 60^(@)`
C. `cos 45^(@)`
D. `cos 38^(@)`

Answer

To solve the expression \( \cos 38^\circ \cos 8^\circ + \sin 38^\circ \sin 8^\circ \), we can use the cosine of the difference of angles formula.

### Step-by-step Solution:

1. **Identify the formula**: We will use the cosine of the sum formula:
\[ \cos(A - B) = \cos A \cos B + \sin A \sin B \]
Here, \( A = 38^\circ \) and \( B = 8^\circ \).

2. **Substitute the values**: Substitute \( A \) and \( B \) into the formula:
\[ \cos(38^\circ - 8^\circ) = \cos 38^\circ \cos 8^\circ + \sin 38^\circ \sin 8^\circ \]

3. **Calculate the angle**: Now, calculate \( 38^\circ - 8^\circ \):
\[ 38^\circ - 8^\circ = 30^\circ \]

4. **Write the final expression**: Thus, we can write:
\[ \cos(30^\circ) = \cos 38^\circ \cos 8^\circ + \sin 38^\circ \sin 8^\circ \]

5. **Evaluate \( \cos(30^\circ) \)**: We know that:
\[ \cos(30^\circ) = \frac{\sqrt{3}}{2} \]

6. **Final answer**: Therefore, the expression \( \cos 38^\circ \cos 8^\circ + \sin 38^\circ \sin 8^\circ \) is equal to:
\[ \frac{\sqrt{3}}{2} \]