Simplify 𝑐𝑜𝑠 2𝑥 𝑐𝑜𝑠 𝑦 + 𝑠𝑖𝑛 2𝑥 𝑠𝑖𝑛 y

The expression given is:

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

This can be simplified using the **product-to-sum** identity for trigonometric functions. The specific identity that applies here is:

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

In this case, \(A = 2x\) and \(B = y\). Therefore:

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

So, the simplified expression is:

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

Would you like more details or have any questions?

Here are 5 related questions you might find interesting:

1. How would you simplify \(\cos(x)\cos(y) + \sin(x)\sin(y)\)?
2. What is the double-angle identity for \(\sin(2x)\)?
3. How can you express \(\cos(3x)\) in terms of \(\cos(x)\) and \(\sin(x)\)?
4. What is the sum-to-product identity for \(\cos(A) + \cos(B)\)?
5. Can you use the identity for \(\cos(A - B)\) in any geometric proofs?

**Tip:** Knowing trigonometric identities can simplify complex expressions and is essential in solving trigonometric equations efficiently.

Learn how to simplify the expression cos(2x) cos(y) + sin(2x) sin(y) using the product-to-sum identity for trigonometric functions. This problem involves trigonometric identities and is suitable for advanced high school students and above.

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