4cos⁴x-5cos²x+sinx sin3x + 3 is equal to

We are tasked with simplifying the expression:

\[
4\cos^4x - 5\cos^2x + \sin x \sin(3x) + 3
\]

### Step 1: Simplify \(\sin x \sin(3x)\)
Using the product-to-sum identity for sine functions, we know that:
\[
\sin A \sin B = \frac{1}{2}[\cos(A - B) - \cos(A + B)]
\]
For \(A = x\) and \(B = 3x\), we have:
\[
\sin x \sin(3x) = \frac{1}{2}[\cos(2x) - \cos(4x)]
\]

So the expression becomes:
\[
4\cos^4x - 5\cos^2x + \frac{1}{2}[\cos(2x) - \cos(4x)] + 3
\]

### Step 2: Rewrite the expression with terms combined
Now substitute the simplified sine product term into the original equation:
\[
4\cos^4x - 5\cos^2x + \frac{1}{2}[\cos(2x) - \cos(4x)] + 3
\]

### Step 3: Analyzing for further simplifications
At this point, the expression can be simplified further depending on specific values for \(x\), but there aren't any general identities to simplify all the terms globally without specific information about \(x\). The structure of the equation suggests it's the final simplified form in terms of trigonometric identities.

The final expression:
\[
4\cos^4x - 5\cos^2x + \frac{1}{2}[\cos(2x) - \cos(4x)] + 3
\]

Would you like me to solve this for a particular value of \(x\) or further explore specific cases?

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### Related questions:
1. How can trigonometric identities be used to simplify complex expressions?
2. What is the product-to-sum identity for sine and cosine?
3. What are some practical applications of trigonometric identities?
4. How do you solve a quartic equation in terms of trigonometric functions?
5. How can we use substitution to simplify \( \cos^4 x \) or \( \cos^2 x \)?

**Tip:** Memorizing key trigonometric identities like the product-to-sum formulas can make solving such problems faster and more intuitive.

4cos⁴x - 5cos²x + sinx sin(3x) + 3 is equal to

Explore how to simplify the trigonometric expression 4cos⁴x - 5cos²x + sinx sin(3x) + 3 using product-to-sum identities and cosine powers. This solution is suitable for high school students in Grades 11-12.

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