The value of the determinant `|{:("cos"^(2)(theta)/(2),"sin"^(2)(theta)/(2)),("sin"^(2)(theta)/(2),"cos"^(2)(theta)/(2)):}|` for all values of `theta`, is

Options

A. 1
B. `cos theta`
C. `sin theta`
D. `cos 2theta`

Explanation

To find the value of the determinant

\[ D = \begin{vmatrix} \frac{\cos^2(\theta)}{2} & \frac{\sin^2(\theta)}{2} \\ \frac{\sin^2(\theta)}{2} & \frac{\cos^2(\theta)}{2} \end{vmatrix} \]

we will use the formula for the determinant of a 2x2 matrix:

\[ \text{If } A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}, \text{ then } |A| = ad - bc \]

### Step 1: Identify the elements of the matrix
Here, we have:
- \( a = \frac{\cos^2(\theta)}{2} \)
- \( b = \frac{\sin^2(\theta)}{2} \)
- \( c = \frac{\sin^2(\theta)}{2} \)
- \( d = \frac{\cos^2(\theta)}{2} \)

### Step 2: Apply the determinant formula
Using the determinant formula:

\[ D = ad - bc \]

Substituting the values we identified:

\[ D = \left(\frac{\cos^2(\theta)}{2}\right) \left(\frac{\cos^2(\theta)}{2}\right) - \left(\frac{\sin^2(\theta)}{2}\right) \left(\frac{\sin^2(\theta)}{2}\right) \]

### Step 3: Simplify the expression
Calculating \( ad \) and \( bc \):

\[ D = \frac{\cos^4(\theta)}{4} - \frac{\sin^4(\theta)}{4} \]

### Step 4: Factor the expression
We can factor the difference of squares:

\[ D = \frac{1}{4} \left(\cos^4(\theta) - \sin^4(\theta)\right) \]

Using the identity \( a^2 - b^2 = (a-b)(a+b) \):

\[ D = \frac{1}{4} \left((\cos^2(\theta) - \sin^2(\theta))(\cos^2(\theta) + \sin^2(\theta))\right) \]

### Step 5: Use the Pythagorean identity
Since \( \cos^2(\theta) + \sin^2(\theta) = 1 \):

\[ D = \frac{1}{4} (\cos^2(\theta) - \sin^2(\theta)) \]

### Step 6: Use the double angle formula
The expression \( \cos^2(\theta) - \sin^2(\theta) \) can be rewritten using the double angle formula:

\[ \cos^2(\theta) - \sin^2(\theta) = \cos(2\theta) \]

Thus, we have:

\[ D = \frac{1}{4} \cos(2\theta) \]

### Final Answer
The value of the determinant for all values of \( \theta \) is:

\[ D = \frac{1}{4} \cos(2\theta) \]