express sin2x*cosy + cos2x*siny as a signle term

To express \(\sin(2x)\cos(y) + \cos(2x)\sin(y)\) as a single term, we can use the **sum-to-product identities**. Specifically, the sum of sines identity is useful here:

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

In this case, let \(A = 2x\) and \(B = y\). Substituting these into the identity:

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

So, the expression \(\sin(2x)\cos(y) + \cos(2x)\sin(y)\) simplifies to:

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

This is the expression as a single term.

Would you like more details on this simplification, or do you have any questions?

Here are 5 related questions for further exploration:

1. How would you simplify \(\sin(x)\cos(y) + \cos(x)\sin(y)\)?
2. Can you express \(\cos(2x)\cos(y) - \sin(2x)\sin(y)\) as a single term?
3. What is the sum-to-product identity for \(\cos(A)\cos(B) + \sin(A)\sin(B)\)?
4. How would you express \(\sin(3x)\cos(y) + \cos(3x)\sin(y)\) as a single term?
5. What is the difference between the product-to-sum and sum-to-product identities?

**Tip:** Familiarizing yourself with trigonometric identities can greatly simplify complex expressions and make solving problems easier.

Learn how to simplify the expression sin(2x)*cos(y) + cos(2x)*sin(y) using trigonometric identities. This detailed explanation covers the application of sum-to-product identities in trigonometry, focusing on the transformation of sums of trigonometric functions into single-term expressions. Suitable for advanced high school students and anyone studying trigonometric identities.

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