`cos^(2)48^(@)-sin^(2)12^(@)=?`

Options

A. `(sqrt(5)-1)/(4)`
B. `(sqrt(5)+1)/(8)`
C. `(sqrt(3)-1)/(4)`
D. `(sqrt(3)+1)/(2sqrt(2))`

Answer

To solve the expression \( \cos^2 48^\circ - \sin^2 12^\circ \), we can use the trigonometric identity that relates the difference of squares of cosine and sine.

### Step-by-Step Solution:

1. **Identify the Formula**:
We use the identity:
\[ \cos^2 A - \sin^2 B = \cos(A + B) \cdot \cos(A - B) \]
Here, let \( A = 48^\circ \) and \( B = 12^\circ \).

2. **Apply the Formula**:
Substitute \( A \) and \( B \) into the identity:
\[ \cos^2 48^\circ - \sin^2 12^\circ = \cos(48^\circ + 12^\circ) \cdot \cos(48^\circ - 12^\circ) \]

3. **Calculate the Angles**:
Now calculate \( 48^\circ + 12^\circ \) and \( 48^\circ - 12^\circ \):
\[ 48^\circ + 12^\circ = 60^\circ \]
\[ 48^\circ - 12^\circ = 36^\circ \]

4. **Substitute Back into the Expression**:
Now we can write:
\[ \cos^2 48^\circ - \sin^2 12^\circ = \cos(60^\circ) \cdot \cos(36^\circ) \]

5. **Evaluate \( \cos(60^\circ) \)**:
We know that:
\[ \cos(60^\circ) = \frac{1}{2} \]

6. **Substitute the Value**:
Therefore, we have:
\[ \cos^2 48^\circ - \sin^2 12^\circ = \frac{1}{2} \cdot \cos(36^\circ) \]

7. **Evaluate \( \cos(36^\circ) \)**:
The value of \( \cos(36^\circ) \) is known to be:
\[ \cos(36^\circ) = \frac{\sqrt{5} + 1}{4} \]

8. **Final Calculation**:
Substitute \( \cos(36^\circ) \) into the expression:
\[ \cos^2 48^\circ - \sin^2 12^\circ = \frac{1}{2} \cdot \frac{\sqrt{5} + 1}{4} = \frac{\sqrt{5} + 1}{8} \]

### Final Answer:
\[ \cos^2 48^\circ - \sin^2 12^\circ = \frac{\sqrt{5} + 1}{8} \]