The value of the determinants `|("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`

Answer

To find the value of the determinant

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

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

\[ D = ad - bc \]

where \( a = \cos^2\left(\frac{\theta}{2}\right) \), \( b = \sin^2\left(\frac{\theta}{2}\right) \), \( c = \sin^2\left(\frac{\theta}{2}\right) \), and \( d = \cos^2\left(\frac{\theta}{2}\right) \).

### Step 1: Calculate the determinant

Substituting the values into the determinant formula:

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

This simplifies to:

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

### Step 2: Apply the difference of squares

The expression \( \cos^4\left(\frac{\theta}{2}\right) - \sin^4\left(\frac{\theta}{2}\right) \) can be factored using the difference of squares:

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

### Step 3: Simplify using trigonometric identities

We know from the Pythagorean identity that:

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

Thus, we can simplify \( D \):

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

So,

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

### Step 4: Use the double angle identity

Using the double angle identity for cosine, we have:

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

Thus, the value of the determinant is:

\[ D = \cos\left(\theta\right) \]

### Final Answer

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

\[ \cos\left(\theta\right) \]

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