Evaluate `|["cos" 15^(@), "sin"15^(@)],["sin" 75^(@), "cos"75^(@)]|`

Answer

To evaluate the determinant of the given matrix:

\[ D = \begin{vmatrix} \cos 15^\circ & \sin 15^\circ \\ \sin 75^\circ & \cos 75^\circ \end{vmatrix} \]

we will follow these steps:

### Step 1: Write the formula for the determinant of a 2x2 matrix
The determinant of a 2x2 matrix

\[ \begin{vmatrix} a & b \\ c & d \end{vmatrix} \]

is given by the formula:

\[ D = ad - bc \]

### Step 2: Identify the elements of the matrix
In our case, we have:
- \( a = \cos 15^\circ \)
- \( b = \sin 15^\circ \)
- \( c = \sin 75^\circ \)
- \( d = \cos 75^\circ \)

### Step 3: Substitute the values into the determinant formula
Now, substituting these values into the determinant formula, we get:

\[ D = \cos 15^\circ \cdot \cos 75^\circ - \sin 15^\circ \cdot \sin 75^\circ \]

### Step 4: Use the cosine addition formula
We can use the cosine addition formula, which states:

\[ \cos(A + B) = \cos A \cdot \cos B - \sin A \cdot \sin B \]

Here, let \( A = 15^\circ \) and \( B = 75^\circ \). Thus, we have:

\[ D = \cos(15^\circ + 75^\circ) \]

### Step 5: Calculate the angle
Calculating the angle:

\[ D = \cos(90^\circ) \]

### Step 6: Evaluate the cosine
The value of \( \cos(90^\circ) \) is:

\[ D = 0 \]

### Final Result
Thus, the value of the determinant is:

\[ \boxed{0} \]