Factor each trigonometric expression.
sin³ α + cos³ α

Verified step by step guidance

1. Recognize that the expression \( \sin^3 \alpha + \cos^3 \alpha \) is a sum of cubes.
2. Recall the sum of cubes formula: \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \).
3. Identify \( a = \sin \alpha \) and \( b = \cos \alpha \) in the expression.
4. Apply the sum of cubes formula: \( \sin^3 \alpha + \cos^3 \alpha = (\sin \alpha + \cos \alpha)(\sin^2 \alpha - \sin \alpha \cos \alpha + \cos^2 \alpha) \).
5. Simplify the second factor using the Pythagorean identity \( \sin^2 \alpha + \cos^2 \alpha = 1 \), resulting in \( 1 - \sin \alpha \cos \alpha \).

Key Concepts

Sum of Cubes Formula

The sum of cubes formula states that a³ + b³ can be factored as (a + b)(a² - ab + b²). This formula is essential for factoring expressions like sin³ α + cos³ α, where a = sin α and b = cos α. Understanding this formula allows us to break down complex trigonometric expressions into simpler components.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variable. Key identities, such as sin² α + cos² α = 1, can be useful when simplifying or manipulating trigonometric expressions. Recognizing these identities helps in transforming and factoring expressions effectively.

Factoring Techniques

Factoring techniques involve rewriting an expression as a product of its factors. This includes recognizing patterns, such as the sum of cubes, and applying algebraic methods to simplify expressions. Mastery of these techniques is crucial for solving trigonometric problems and simplifying complex expressions.