`cos^2 48^0-sin^2 12^0` is equal to

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 for the difference of squares.

### Step-by-Step Solution:

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

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

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

4. **Substitute the Values**:
Now substitute these values back into the equation:
\[ \cos^2 48^\circ - \sin^2 12^\circ = \cos(60^\circ) \cos(36^\circ) \]

5. **Evaluate the Cosine Values**:
We know that:
\[ \cos(60^\circ) = \frac{1}{2} \]
For \( \cos(36^\circ) \), we can use the known value:
\[ \cos(36^\circ) = \frac{\sqrt{5} + 1}{4} \]

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

7. **Simplify the Expression**:
Multiply the fractions:
\[ = \frac{\sqrt{5} + 1}{8} \]

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