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

We are asked to simplify the expression:

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

### Step 1: Simplify the trigonometric terms

#### Use product-to-sum identities
For the product \(\sin(x) \sin(3x)\), we can use the product-to-sum identity:

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

Substituting \(A = x\) and \(B = 3x\), we get:

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

Since \(\cos(-2x) = \cos(2x)\), this simplifies to:

\[
\sin(x) \sin(3x) = \frac{1}{2} [\cos(2x) - \cos(4x)]
\]

### Step 2: Substitute into the original expression
Now, substitute this result back into the original expression:

\[
4\cos(4x) - 5\cos(2x) + \frac{1}{2} [\cos(2x) - \cos(4x)] + 3
\]

### Step 3: Combine like terms
Group the \(\cos(4x)\) and \(\cos(2x)\) terms:

\[
(4\cos(4x) - \frac{1}{2} \cos(4x)) + (-5\cos(2x) + \frac{1}{2} \cos(2x)) + 3
\]

Simplify each group:

\[
(4 - \frac{1}{2}) \cos(4x) = \frac{7}{2} \cos(4x)
\]
\[
(-5 + \frac{1}{2}) \cos(2x) = -\frac{9}{2} \cos(2x)
\]

Thus, the expression simplifies to:

\[
\frac{7}{2} \cos(4x) - \frac{9}{2} \cos(2x) + 3
\]

This is the simplified form of the expression.

Would you like more details, or do you have any further questions?

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Here are some related questions you can explore:
1. How do you apply sum and difference formulas for sine and cosine?
2. What are product-to-sum identities, and how are they derived?
3. Can this expression be solved for a specific value of \(x\)?
4. What are the general solutions for trigonometric equations involving \(\cos(4x)\) and \(\cos(2x)\)?
5. How do phase shifts affect trigonometric functions?

**Tip**: When working with trigonometric identities, it's often helpful to rewrite products as sums to simplify expressions.

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

Discover how to simplify the trigonometric expression 4cos(4x) - 5cos(2x) + sin(x)sin(3x) + 3 using product-to-sum identities and trigonometric simplifications. This detailed guide walks through each step and explains key trigonometric concepts for high school students.

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