If the distance between the points `P (a cos 48^@, 0) and Q(0, a cos 12^@)` is `d,` then `d^2-a^2=`

Options

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

Answer

To solve the problem, we need to find the expression for \( d^2 - a^2 \) given the points \( P(a \cos 48^\circ, 0) \) and \( Q(0, a \cos 12^\circ) \).

### Step-by-Step Solution:

1. **Identify the Points**:
- The coordinates of point \( P \) are \( (a \cos 48^\circ, 0) \).
- The coordinates of point \( Q \) are \( (0, a \cos 12^\circ) \).

2. **Use the Distance Formula**:
The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Here, substituting the coordinates of \( P \) and \( Q \):
\[ d = \sqrt{(0 - a \cos 48^\circ)^2 + (a \cos 12^\circ - 0)^2} \]

3. **Simplify the Expression**:
\[ d = \sqrt{(-a \cos 48^\circ)^2 + (a \cos 12^\circ)^2} \]
This simplifies to:
\[ d = \sqrt{(a^2 \cos^2 48^\circ) + (a^2 \cos^2 12^\circ)} \]

4. **Factor Out \( a^2 \)**:
\[ d = \sqrt{a^2 (\cos^2 48^\circ + \cos^2 12^\circ)} \]
\[ d = a \sqrt{\cos^2 48^\circ + \cos^2 12^\circ} \]

5. **Square Both Sides**:
\[ d^2 = a^2 (\cos^2 48^\circ + \cos^2 12^\circ) \]

6. **Find \( d^2 - a^2 \)**:
\[ d^2 - a^2 = a^2 (\cos^2 48^\circ + \cos^2 12^\circ) - a^2 \]
\[ d^2 - a^2 = a^2 (\cos^2 48^\circ + \cos^2 12^\circ - 1) \]

7. **Use the Identity**:
We know that \( \cos^2 \theta + \sin^2 \theta = 1 \). Therefore:
\[ \cos^2 48^\circ + \cos^2 12^\circ - 1 = \cos^2 48^\circ + \cos^2 12^\circ - \sin^2 12^\circ - \sin^2 48^\circ \]

Using the cosine double angle identity:
\[ \cos^2 48^\circ + \cos^2 12^\circ = \frac{1 + \cos 96^\circ}{2} + \frac{1 + \cos 24^\circ}{2} \]
This leads to:
\[ d^2 - a^2 = a^2 \left( \frac{2 + \cos 96^\circ + \cos 24^\circ}{2} - 1 \right) \]

8. **Final Expression**:
After simplification, we find:
\[ d^2 - a^2 = a^2 \cdot \frac{\cos 96^\circ + \cos 24^\circ}{2} \]

### Final Answer:
\[ d^2 - a^2 = a^2 \cdot \frac{\cos 96^\circ + \cos 24^\circ}{2} \]