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)/(5)`
D. `(sqrt(5)+1)/(2sqrt(2))`

Explanation

To find 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. **Use the identity**:
We know that:
\[ \cos^2 A - \sin^2 B = \cos(A + B) \cdot \sin(A - B) \]
Here, let \( A = 48^\circ \) and \( B = 12^\circ \).

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

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 the angles**:
Now we substitute these values back into the equation:
\[ \cos^2 48^\circ - \sin^2 12^\circ = \cos 60^\circ \cdot \sin 36^\circ \]

5. **Find the values of \( \cos 60^\circ \) and \( \sin 36^\circ \)**:
We know that:
\[ \cos 60^\circ = \frac{1}{2} \]
For \( \sin 36^\circ \), we can use the known value:
\[ \sin 36^\circ = \frac{\sqrt{5} - 1}{4} \]

6. **Multiply the values**:
Now, substitute these values into the equation:
\[ \cos^2 48^\circ - \sin^2 12^\circ = \frac{1}{2} \cdot \frac{\sqrt{5} - 1}{4} \]
\[ = \frac{\sqrt{5} - 1}{8} \]

### Final Result:
Thus, the value of \( \cos^2 48^\circ - \sin^2 12^\circ \) is:
\[ \frac{\sqrt{5} - 1}{8} \]