1. Simplifying $\sin(4x) \cos(x)$

To simplify this expression using the Product-to-Sum formula, we use the following identity:

$\sin(A) \cos(B) = \frac{1}{2} \left[ \sin(A + B) + \sin(A - B) \right]$

For $A = 4x$ and $B = x$, the expression becomes:

$\sin(4x) \cos(x) = \frac{1}{2} \left[ \sin(4x + x) + \sin(4x - x) \right]$

Simplifying the terms inside the sine functions:

$= \frac{1}{2} \left[ \sin(5x) + \sin(3x) \right]$

Thus, the simplified form is:

$\sin(4x) \cos(x) = \frac{1}{2} \left[ \sin(5x) + \sin(3x) \right]$

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2. Simplifying $\cos(5x) \cos(2x)$

For this expression, we use the Product-to-Sum formula for cosines:

$\cos(A) \cos(B) = \frac{1}{2} \left[ \cos(A + B) + \cos(A - B) \right]$

Here, $A = 5x$ and $B = 2x$. Applying the formula:

$\cos(5x) \cos(2x) = \frac{1}{2} \left[ \cos(5x + 2x) + \cos(5x - 2x) \right]$

Simplifying the terms inside the cosine functions:

$= \frac{1}{2} \left[ \cos(7x) + \cos(3x) \right]$

Thus, the simplified form is:

$\cos(5x) \cos(2x) = \frac{1}{2} \left[ \cos(7x) + \cos(3x) \right]$

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Would you like further details or explanations on these steps?

Related questions:

1. How do the Product-to-Sum formulas apply to other trigonometric functions, like $\sin(x)\sin(y)$ or $\cos(x)\sin(y)$?
2. Can the Product-to-Sum formulas be used to simplify integrals involving trigonometric functions?
3. How would you use the Sum-to-Product formulas to simplify expressions?
4. What is the geometric interpretation of the Product-to-Sum identities?
5. How can you derive the Product-to-Sum formulas from the angle addition formulas?

Tip: Remember that the Product-to-Sum formulas are especially useful for transforming products of trigonometric functions into sums, which can simplify integration and other mathematical operations.