If `A=[[1, tan((theta)/(2))], [-tan((theta)/(2)), 1]] and AB=I`, then B=

Options

A. `cos^(2)((theta)/(2))I`
B. `cos^(2)((theta)/(2))A`
C. `cos^(2)((theta)/(2))A^(T)`
D. `-cos^(2)((theta)/(2))A^(T)`

Answer

To solve the problem, we need to find the matrix \( B \) such that \( AB = I \), where \( A = \begin{bmatrix} 1 & \tan\left(\frac{\theta}{2}\right) \\ -\tan\left(\frac{\theta}{2}\right) & 1 \end{bmatrix} \) and \( I \) is the identity matrix.

### Step 1: Find the determinant of matrix \( A \)

The determinant of a \( 2 \times 2 \) matrix \( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \) is calculated as \( ad - bc \).

For matrix \( A \):
\[ \text{det}(A) = 1 \cdot 1 - \left(-\tan\left(\frac{\theta}{2}\right) \cdot \tan\left(\frac{\theta}{2}\right)\right) = 1 + \tan^2\left(\frac{\theta}{2}\right) \]

Using the identity \( 1 + \tan^2 x = \sec^2 x \):
\[ \text{det}(A) = \sec^2\left(\frac{\theta}{2}\right) \]

### Step 2: Find the adjoint of matrix \( A \)

The adjoint of a \( 2 \times 2 \) matrix \( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \) is given by \( \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \).

For matrix \( A \):
\[ \text{adj}(A) = \begin{bmatrix} 1 & -\tan\left(\frac{\theta}{2}\right) \\ \tan\left(\frac{\theta}{2}\right) & 1 \end{bmatrix} \]

### Step 3: Calculate the inverse of matrix \( A \)

The inverse of a matrix \( A \) is given by the formula:
\[ A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A) \]

Substituting the values we found:
\[ A^{-1} = \frac{1}{\sec^2\left(\frac{\theta}{2}\right)} \cdot \begin{bmatrix} 1 & -\tan\left(\frac{\theta}{2}\right) \\ \tan\left(\frac{\theta}{2}\right) & 1 \end{bmatrix} \]
\[ = \cos^2\left(\frac{\theta}{2}\right) \cdot \begin{bmatrix} 1 & -\tan\left(\frac{\theta}{2}\right) \\ \tan\left(\frac{\theta}{2}\right) & 1 \end{bmatrix} \]

### Step 4: Set \( B = A^{-1} \)

Since \( AB = I \), we have:
\[ B = A^{-1} = \cos^2\left(\frac{\theta}{2}\right) \cdot \begin{bmatrix} 1 & -\tan\left(\frac{\theta}{2}\right) \\ \tan\left(\frac{\theta}{2}\right) & 1 \end{bmatrix} \]

### Step 5: Identify the correct option

Now we need to express \( B \) in terms of the options given. The matrix \( A \) can be expressed as:
\[ A = \begin{bmatrix} 1 & \tan\left(\frac{\theta}{2}\right) \\ -\tan\left(\frac{\theta}{2}\right) & 1 \end{bmatrix} \]

Thus, we can express \( B \) as:
\[ B = \cos^2\left(\frac{\theta}{2}\right) \cdot A^T \]

### Final Answer

The correct option for \( B \) is:
\[ B = \cos^2\left(\frac{\theta}{2}\right) \cdot A^T \]