To linearize $\cos^3x$, we use trigonometric identities to express the term in a linear form with respect to basic trigonometric functions. Here's the process:

Step 1: Use the power reduction formula

The power reduction formula for cosine is: $\cos^2x = \frac{1 + \cos(2x)}{2}.$

Therefore: $\cos^3x = \cos x \cdot \cos^2x = \cos x \cdot \frac{1 + \cos(2x)}{2}.$

Step 2: Expand the expression

Substituting the formula: $\cos^3x = \frac{\cos x}{2} + \frac{\cos x \cdot \cos(2x)}{2}.$

Step 3: Simplify the second term

Using the product-to-sum identity: [ \cos x \cdot \cos(2x) = \frac{1}{2} \left[\cos(x - 2x) + \cos(x + 2x)\right] = \frac{1}{2} \left[\cos(-x) + \cos(3x)\right]. ] Since $\cos(-x) = \cos(x)$: $\cos x \cdot \cos(2x) = \frac{1}{2} \left[\cos(x) + \cos(3x)\right].$

Step 4: Substitute back

Now, substitute this into the expression: $\cos^3x = \frac{\cos x}{2} + \frac{1}{2} \cdot \frac{\cos(x) + \cos(3x)}{2}.$

Simplify: $\cos^3x = \frac{\cos x}{2} + \frac{\cos x}{4} + \frac{\cos(3x)}{4}.$

Combine like terms: $\cos^3x = \frac{2\cos x}{4} + \frac{\cos x}{4} + \frac{\cos(3x)}{4} = \frac{3\cos x}{4} + \frac{\cos(3x)}{4}.$

Final Linearized Form:

$\cos^3x = \frac{3}{4} \cos x + \frac{1}{4} \cos(3x).$

Let me know if you want further explanation or related examples!

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Questions:

1. What is the purpose of linearizing trigonometric expressions?
2. How do power-reduction formulas simplify expressions like $\cos^n x$?
3. Can $\cos^n x$ be linearized for any $n$? How does the process vary with higher powers?
4. How are product-to-sum formulas applied to trigonometric products?
5. What are some applications of linearized trigonometric expressions in calculus or physics?

Tip:

When working with higher powers of trigonometric functions, always check for symmetry and reduction formulas—they simplify complex expressions significantly.