Evaluate : `|{:(costheta,-sintheta),(sintheta,costheta):}|`

Answer

To evaluate the determinant \( D = \begin{vmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{vmatrix} \), we will follow the steps below:

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

### Step 2: Apply the formula for a 2x2 determinant
For a 2x2 matrix \( \begin{vmatrix} a & b \\ c & d \end{vmatrix} \), the determinant is calculated as:
\[ D = ad - bc \]
In our case:
- \( a = \cos \theta \)
- \( b = -\sin \theta \)
- \( c = \sin \theta \)
- \( d = \cos \theta \)

### Step 3: Substitute the values into the determinant formula
Now substituting the values into the formula:
\[ D = (\cos \theta)(\cos \theta) - (-\sin \theta)(\sin \theta) \]

### Step 4: Simplify the expression
This simplifies to:
\[ D = \cos^2 \theta + \sin^2 \theta \]

### Step 5: Use the Pythagorean identity
We know from the Pythagorean identity that:
\[ \cos^2 \theta + \sin^2 \theta = 1 \]

### Step 6: Conclude the evaluation
Thus, we find that:
\[ D = 1 \]

### Final Answer
The value of the determinant is:
\[ \boxed{1} \]