Expand `| (cosx, -sinx),(sinx , cosx)|`

Answer

To expand the determinant \( | ( \cos x, -\sin x ), ( \sin x, \cos x ) | \), we can follow these steps:

### Step 1: Write the determinant
We start with the determinant:
\[ \Delta = \begin{vmatrix} \cos x & -\sin x \\ \sin x & \cos x \end{vmatrix} \]

### Step 2: Apply the determinant formula
The formula for the determinant of a 2x2 matrix \( \begin{vmatrix} a & b \\ c & d \end{vmatrix} \) is given by \( ad - bc \). Here, we have:
- \( a = \cos x \)
- \( b = -\sin x \)
- \( c = \sin x \)
- \( d = \cos x \)

So, we can calculate the determinant as follows:
\[ \Delta = (\cos x)(\cos x) - (-\sin x)(\sin x) \]

### Step 3: Simplify the expression
Now, we simplify the expression:
\[ \Delta = \cos^2 x + \sin^2 x \]

### Step 4: Use the Pythagorean identity
We know from the Pythagorean identity that:
\[ \cos^2 x + \sin^2 x = 1 \]

### Final Result
Thus, the value of the determinant is:
\[ \Delta = 1 \]