The expression \( 2\cos(2t)\cos(4t) \) can be simplified using the product-to-sum identities. The relevant identity is:

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

In your case, \( A = 2t \) and \( B = 4t \). Plugging these into the identity gives:

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

Simplifying inside the cosine functions:

\[
\cos(2t)\cos(4t) = \frac{1}{2}\left[\cos(-2t) + \cos(6t)\right]
\]

Since \( \cos(-x) = \cos(x) \):

\[
\cos(2t)\cos(4t) = \frac{1}{2}\left[\cos(2t) + \cos(6t)\right]
\]

Multiplying by 2 to account for the original coefficient:

\[
2\cos(2t)\cos(4t) = \cos(2t) + \cos(6t)
\]

So the simplified expression is:

\[
\boxed{\cos(2t) + \cos(6t)}
\]

Would you like more details or have any questions about this process? 

Here are some related questions you might consider:

1. How do the product-to-sum identities help in simplifying trigonometric expressions?
2. What other product-to-sum identities are there in trigonometry?
3. How can we apply this identity in solving integrals involving trigonometric functions?
4. Can you simplify the expression \( \sin(3t)\sin(5t) \) using a similar identity?
5. How does this identity apply in Fourier series or signal processing?

**Tip:** Understanding the product-to-sum identities is particularly useful in simplifying expressions involving products of trigonometric functions, which can appear frequently in calculus and physics problems.

Learn how to simplify the expression 2cos(2t)cos(4t) step-by-step using trigonometric identities. This problem covers the application of product-to-sum identities in trigonometry, useful for advanced high school students and calculus enthusiasts.

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