The value of `"cos"^(2)48^(@)-"sin"^(2)12^(@)` is

Options

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

Answer

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

### Step-by-Step Solution:

1. **Identify the expression**: We need to evaluate \( \cos^2 48^\circ - \sin^2 12^\circ \).

2. **Use the identity**: We can use the identity \( \cos^2 A - \sin^2 B = \cos(A + B) \cdot \cos(A - B) \). Here, let \( A = 48^\circ \) and \( B = 12^\circ \).

3. **Calculate \( A + B \) and \( A - B \)**:
- \( A + B = 48^\circ + 12^\circ = 60^\circ \)
- \( A - B = 48^\circ - 12^\circ = 36^\circ \)

4. **Substitute into the identity**:
\[ \cos^2 48^\circ - \sin^2 12^\circ = \cos(60^\circ) \cdot \cos(36^\circ) \]

5. **Find the values of \( \cos(60^\circ) \) and \( \cos(36^\circ) \)**:
- \( \cos(60^\circ) = \frac{1}{2} \)
- \( \cos(36^\circ) = \frac{\sqrt{5} + 1}{4} \) (This is a known value)

6. **Multiply the results**:
\[ \cos^2 48^\circ - \sin^2 12^\circ = \frac{1}{2} \cdot \frac{\sqrt{5} + 1}{4} \]
\[ = \frac{\sqrt{5} + 1}{8} \]

7. **Final result**: Therefore, the value of \( \cos^2 48^\circ - \sin^2 12^\circ \) is \( \frac{\sqrt{5} + 1}{8} \).

### Conclusion:
The answer is \( \frac{\sqrt{5} + 1}{8} \).