`A=[[costheta,isintheta],[isintheta,costheta]]` and `A^5=[[a,d],[b,c]]`. Then which one is correct.

Options

A. `a^2-d^2=0`
B. `a^2+c^2=1`
C. `a^2-c^2=3/2`
D. `a^2-b^2=1`

Answer

To solve the problem, we need to analyze the given matrix \( A \) and its powers. The matrix \( A \) is defined as:

\[ A = \begin{bmatrix} \cos \theta & i \sin \theta \\ i \sin \theta & \cos \theta \end{bmatrix} \]

According to the theorem mentioned in the video, for a matrix of this form, the \( n \)-th power of \( A \) can be expressed as:

\[ A^n = \begin{bmatrix} \cos(n\theta) & i \sin(n\theta) \\ i \sin(n\theta) & \cos(n\theta) \end{bmatrix} \]

Given that \( A^5 = \begin{bmatrix} a & d \\ b & c \end{bmatrix} \), we can substitute \( n = 5 \) into the theorem:

\[ A^5 = \begin{bmatrix} \cos(5\theta) & i \sin(5\theta) \\ i \sin(5\theta) & \cos(5\theta) \end{bmatrix} \]

From this, we can identify:

- \( a = \cos(5\theta) \)
- \( d = i \sin(5\theta) \)
- \( b = i \sin(5\theta) \)
- \( c = \cos(5\theta) \)

Now, we need to check the options provided in the problem statement. We will evaluate the following expressions:

1. \( a^2 - d^2 \)
2. \( a^2 - b^2 - d^2 \)
3. \( a^2 + c^2 \)
4. \( a^2 - b^2 \)

### Step 1: Calculate \( a^2 - d^2 \)

\[ a^2 = \cos^2(5\theta) \]
\[ d^2 = (i \sin(5\theta))^2 = -\sin^2(5\theta) \]
Thus, we have:

\[ a^2 - d^2 = \cos^2(5\theta) - (-\sin^2(5\theta)) = \cos^2(5\theta) + \sin^2(5\theta) = 1 \]

### Step 2: Calculate \( a^2 - b^2 - d^2 \)

\[ b^2 = (i \sin(5\theta))^2 = -\sin^2(5\theta) \]
So,

\[ a^2 - b^2 - d^2 = \cos^2(5\theta) - (-\sin^2(5\theta)) - (-\sin^2(5\theta)) \]
This simplifies to:

\[ \cos^2(5\theta) + \sin^2(5\theta) + \sin^2(5\theta) = 1 + \sin^2(5\theta) \]

This expression is not equal to 1 for all values of \( \theta \).

### Step 3: Calculate \( a^2 + c^2 \)

Since \( c = \cos(5\theta) \):

\[ a^2 + c^2 = \cos^2(5\theta) + \cos^2(5\theta) = 2\cos^2(5\theta) \]

This is not equal to 1 unless \( \cos(5\theta) = \frac{1}{\sqrt{2}} \).

### Step 4: Calculate \( a^2 - b^2 \)

\[ a^2 - b^2 = \cos^2(5\theta) - (-\sin^2(5\theta)) = \cos^2(5\theta) + \sin^2(5\theta) = 1 \]

### Conclusion

From the calculations, we find:

- \( a^2 - d^2 = 1 \) (True)
- \( a^2 - b^2 - d^2 = 1 + \sin^2(5\theta) \) (Not always true)
- \( a^2 + c^2 = 2\cos^2(5\theta) \) (Not always true)
- \( a^2 - b^2 = 1 \) (True)

Thus, the correct options are those that yield \( 1 \).

### Final Answer

The correct option is:

- \( a^2 - d^2 = 1 \)
- \( a^2 - b^2 = 1 \)