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The given expression \( 4 \cos^4(x) - 5 \cos^2(x) + \sin(x) \sin(3x) + 3 \) simplifies to:

\[
\cos(2x)^2 - \frac{1}{2} \cos(4x) + \frac{3}{2}
\]

Now, for single-digit integer values of \( x \), this can be evaluated further based on specific values of \( x \). If you want to check specific values, let me know, or I can proceed with evaluating it further.

Would you like to see the detailed steps or have any other questions? Here are five related questions for you to consider:
1. How do you use double-angle identities for trigonometric simplification?
2. What is the relationship between powers of cosine and multiple-angle formulas?
3. How can trigonometric products like \( \sin(x) \sin(3x) \) be rewritten using sum-to-product identities?
4. How can you differentiate the expression you just simplified?
5. What values of \( x \) make \( \cos(2x)^2 - \frac{1}{2} \cos(4x) + \frac{3}{2} = 0 \)?

**Tip**: Trigonometric identities can often help simplify complex expressions and make them easier to evaluate or solve!

4cos^4(x) - 5cos^2(x) + sin(x)sin(3x) + 3 is equal to?

Learn how to simplify the trigonometric expression 4cos^4(x) - 5cos^2(x) + sin(x)sin(3x) + 3 using double-angle identities and sum-to-product formulas. Explore detailed steps for high school students in Grades 11-12.

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