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The new image you uploaded shows another mathematical problem involving a determinant. Let's read and solve it.

**Given Problem:**

The problem is to calculate the determinant of a 2x2 matrix:

\[
\begin{bmatrix}
\sin^2 x & 1 \\
1 & \cos^2 x
\end{bmatrix}
\]

**Solution:**

To find the determinant of the 2x2 matrix \(\begin{bmatrix} a & b \\ c & d \end{bmatrix}\), use the formula:

\[
\text{det} = ad - bc
\]

Applying this formula to the given matrix:

\[
\text{det} = (\sin^2 x)(\cos^2 x) - (1)(1)
\]

\[
\text{det} = \sin^2 x \cos^2 x - 1
\]

**Simplification Using Trigonometric Identities:**

We know from trigonometric identities that:

\[
\sin^2 x + \cos^2 x = 1
\]

This identity doesn't directly simplify \(\sin^2 x \cos^2 x\), so let's leave the expression as it is.

Thus, the determinant is:

\[
\text{det} = \sin^2 x \cos^2 x - 1
\]

Therefore, the answer to the given problem is:

\[
\sin^2 x \cos^2 x - 1
\]

Would you like more details or have any further questions?

Here are five related questions to expand your understanding:

1. How can you use trigonometric identities to further simplify expressions involving \(\sin^2 x\) and \(\cos^2 x\)?
2. What happens to the determinant of this matrix when \(x = \frac{\pi}{4}\)?
3. How do you interpret the determinant in terms of linear transformations?
4. How does the determinant change if we replace \(1\) with another constant \(k\) in the matrix?
5. Can we use row or column operations to make calculating determinants easier in certain cases?

**Tip:** For matrices involving trigonometric functions, always consider using trigonometric identities to simplify the expressions when possible.

Learn how to calculate the determinant of a 2x2 matrix involving trigonometric functions. This problem explores matrix algebra, determinants, and trigonometric identities. Suitable for advanced high school students.

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