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The problem asks to simplify the following \( 3 \times 3 \) matrix expression:

\[
\begin{pmatrix}
1 & x & y \\
0 & \cos x & \sin y \\
0 & \sin x & \cos y
\end{pmatrix}
\]

The given answer choices are:

a) \( \cos(x + y) \)

b) \( \cos x \cos y + \sin x \sin y \)

c) \( 1 - \cos x \sin y \)

d) \( 2 \sin x + \cos y \)

### Step-by-step Simplification:
This problem likely involves multiplying vectors or combining matrix elements into a simpler trigonometric form. We observe that the second and third rows contain trigonometric functions, which suggests the use of trigonometric identities. Specifically, the expression \( \cos x \cos y + \sin x \sin y \) is the formula for \( \cos(x + y) \):

\[
\cos(x + y) = \cos x \cos y + \sin x \sin y
\]

Thus, based on this trigonometric identity, the expression simplifies to:

\[
\boxed{b) \ \cos x \cos y + \sin x \sin y}
\]

This matches with choice b.

Let me know if you would like more detailed steps on how to reach this answer or have any questions!

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**5 related questions:**
1. What is the significance of trigonometric identities in simplifying matrices?
2. How does matrix multiplication relate to trigonometric simplifications?
3. Can we generalize the trigonometric identity \( \cos(x + y) \) for other operations?
4. What happens if one of the angles in the matrix is zero?
5. How do different types of matrices (e.g., diagonal, identity) simplify under trigonometric operations?

**Tip:** Familiarity with common trigonometric identities, such as sum and difference formulas, can simplify matrix operations significantly.

Simplify the 3x3 matrix: [[1, x, y], [0, cos(x), sin(y)], [0, sin(x), cos(y)]]

Explore how to simplify a 3x3 matrix containing trigonometric functions of x and y, using the cosine of sum identity. This advanced topic is suitable for college-level students studying matrix algebra and trigonometry.

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