What is the value of `|{:(cos 15^(@),sin 15^(@)),(cos 45^(@),sin 45^(@)):}|xx|{:(cos 45^(@),cos 15^(@)),(sin 45^(@),sin 15^(@)):}|`?

Options

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

Explanation

To solve the given problem, we need to evaluate the expression:

\[ | \begin{pmatrix} \cos 15^\circ & \sin 15^\circ \\ \cos 45^\circ & \sin 45^\circ \end{pmatrix} | \times | \begin{pmatrix} \cos 45^\circ & \cos 15^\circ \\ \sin 45^\circ & \sin 15^\circ \end{pmatrix} | \]

### Step 1: Calculate the first determinant

The first determinant is given by:

\[ D_1 = \begin{vmatrix} \cos 15^\circ & \sin 15^\circ \\ \cos 45^\circ & \sin 45^\circ \end{vmatrix} \]

Using the formula for the determinant of a 2x2 matrix:

\[ D_1 = (\cos 15^\circ)(\sin 45^\circ) - (\sin 15^\circ)(\cos 45^\circ) \]

### Step 2: Substitute the values of sine and cosine

We know that:

\[ \sin 45^\circ = \frac{\sqrt{2}}{2} \quad \text{and} \quad \cos 45^\circ = \frac{\sqrt{2}}{2} \]

Substituting these values into the determinant:

\[ D_1 = \cos 15^\circ \cdot \frac{\sqrt{2}}{2} - \sin 15^\circ \cdot \frac{\sqrt{2}}{2} \]

Factoring out \(\frac{\sqrt{2}}{2}\):

\[ D_1 = \frac{\sqrt{2}}{2} (\cos 15^\circ - \sin 15^\circ) \]

### Step 3: Calculate the second determinant

The second determinant is given by:

\[ D_2 = \begin{vmatrix} \cos 45^\circ & \cos 15^\circ \\ \sin 45^\circ & \sin 15^\circ \end{vmatrix} \]

Using the determinant formula again:

\[ D_2 = (\cos 45^\circ)(\sin 15^\circ) - (\sin 45^\circ)(\cos 15^\circ) \]

Substituting the values of sine and cosine:

\[ D_2 = \frac{\sqrt{2}}{2} \cdot \sin 15^\circ - \frac{\sqrt{2}}{2} \cdot \cos 15^\circ \]

Factoring out \(\frac{\sqrt{2}}{2}\):

\[ D_2 = \frac{\sqrt{2}}{2} (\sin 15^\circ - \cos 15^\circ) \]

### Step 4: Multiply the two determinants

Now we can find the product of the two determinants:

\[ D = D_1 \times D_2 = \left( \frac{\sqrt{2}}{2} (\cos 15^\circ - \sin 15^\circ) \right) \times \left( \frac{\sqrt{2}}{2} (\sin 15^\circ - \cos 15^\circ) \right) \]

This simplifies to:

\[ D = \frac{2}{4} (\cos 15^\circ - \sin 15^\circ)(\sin 15^\circ - \cos 15^\circ) \]

### Step 5: Simplify the expression

Notice that \((\sin 15^\circ - \cos 15^\circ) = -(\cos 15^\circ - \sin 15^\circ)\). Thus, we can rewrite the product:

\[ D = \frac{1}{2} (\cos 15^\circ - \sin 15^\circ)(-\cos 15^\circ + \sin 15^\circ) \]

This leads to:

\[ D = -\frac{1}{2} (\cos 15^\circ - \sin 15^\circ)^2 \]

### Final Result

Now we can evaluate \((\cos 15^\circ - \sin 15^\circ)^2\) if needed, but the main result is:

\[ D = -\frac{1}{2} (\cos 15^\circ - \sin 15^\circ)^2 \]