Simplify 𝑐𝑜𝑠 2𝑥 𝑐𝑜𝑠 𝑦 + 𝑠𝑖𝑛 2𝑥 𝑠𝑖𝑛 y

To simplify the expression \( \cos(2x) \cos(y) + \sin(2x) \sin(y) \), we can use the **sum-to-product** identity, specifically:

\[
\cos(A) \cos(B) + \sin(A) \sin(B) = \cos(A - B)
\]

In this case, \( A = 2x \) and \( B = y \). So, the expression simplifies to:

\[
\cos(2x - y)
\]

Thus, the simplified form of the expression is:

\[
\cos(2x - y)
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. How would you simplify \( \cos(2x) \cos(y) - \sin(2x) \sin(y) \)?
2. What is the sum-to-product identity for \( \sin(A) \cos(B) + \cos(A) \sin(B) \)?
3. How do you simplify \( \cos(3x) \cos(y) + \sin(3x) \sin(y) \)?
4. Can you express \( \cos(2x) \cos(y) + \sin(2x) \sin(y) \) using complex exponentials?
5. How does the double-angle identity for cosine relate to the sum-to-product identity used here?

**Tip:** Remember that identities like sum-to-product are powerful tools for simplifying trigonometric expressions!

Explore how to simplify the trigonometric expression cos(2x) cos(y) + sin(2x) sin(y) using the sum-to-product identity. This solution provides a detailed step-by-step explanation and is suitable for advanced learners in trigonometry.

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