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**:

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

In this case, let \( A = 2x \) and \( B = y \). Applying the identity, we have:

\[
\cos(2x) \cos(y) + \sin(2x) \sin(y) = \cos(2x - y)
\]

So, the simplified expression is:

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

Would you like to see more details, or do you have any further questions?

Here are 5 related questions you might find interesting:
1. How would you simplify \( \cos(2x) \cos(y) - \sin(2x) \sin(y) \)?
2. What is the sum-to-product identity for \( \cos(A) \cos(B) - \sin(A) \sin(B) \)?
3. How would you express \( \cos(2x - y) \) in terms of \( \cos(2x) \) and \( \sin(y) \)?
4. What are the other sum-to-product identities?
5. How does this identity help in solving trigonometric equations?

**Tip:** Knowing trigonometric identities can greatly simplify complex expressions, making problem-solving faster and more efficient.

Learn how to simplify the expression \( \cos(2x) \cos(y) + \sin(2x) \sin(y) \) using trigonometric identities. This problem covers key concepts in trigonometry, including sum-to-product identities. Suitable for high school students studying trigonometry.

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