Find the value of `cos28^(@).cos32^(@)-sin28^(@).sin32^(@)`:

Options

A. 1
B. `1/2`
C. `1/3`
D. can't be determined

Answer

To solve the expression \( \cos 28^\circ \cdot \cos 32^\circ - \sin 28^\circ \cdot \sin 32^\circ \), we can use the cosine addition formula. The cosine addition formula states that:

\[ \cos(A + B) = \cos A \cdot \cos B - \sin A \cdot \sin B \]

### Step 1: Identify A and B
In our case, we can identify:
- \( A = 28^\circ \)
- \( B = 32^\circ \)

### Step 2: Apply the Cosine Addition Formula
Using the cosine addition formula, we can rewrite the expression:

\[ \cos 28^\circ \cdot \cos 32^\circ - \sin 28^\circ \cdot \sin 32^\circ = \cos(28^\circ + 32^\circ) \]

### Step 3: Calculate the Angle
Now, we calculate \( 28^\circ + 32^\circ \):

\[ 28^\circ + 32^\circ = 60^\circ \]

### Step 4: Find the Cosine Value
Now we need to find \( \cos 60^\circ \):

\[ \cos 60^\circ = \frac{1}{2} \]

### Conclusion
Thus, the value of the expression \( \cos 28^\circ \cdot \cos 32^\circ - \sin 28^\circ \cdot \sin 32^\circ \) is:

\[ \frac{1}{2} \]