Verify that each equation is an identity.
sin³ θ + cos³ θ = (cos θ + sin θ) (1 - cos θ sin θ)

Verified step by step guidance

1. Start by recalling the algebraic identity for the sum of cubes: \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\). Here, let \(a = \sin \theta\) and \(b = \cos \theta\).
2. Rewrite the left side using the sum of cubes formula: \(\sin^3 \theta + \cos^3 \theta = (\sin \theta + \cos \theta)(\sin^2 \theta - \sin \theta \cos \theta + \cos^2 \theta)\).
3. Use the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\) to simplify the expression inside the parentheses: replace \(\sin^2 \theta + \cos^2 \theta\) with 1.
4. After substitution, the expression becomes \((\sin \theta + \cos \theta)(1 - \sin \theta \cos \theta)\), which matches the right side of the given equation.
5. Conclude that since both sides simplify to the same expression, the given equation is an identity.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. Verifying an identity means showing both sides simplify to the same expression, often using known formulas like Pythagorean identities or algebraic manipulations.

Algebraic Factorization

Algebraic factorization involves rewriting expressions as products of simpler factors. Recognizing patterns such as sum of cubes, a³ + b³ = (a + b)(a² - ab + b²), helps in breaking down complex trigonometric expressions to verify identities efficiently.

Pythagorean Identity

The Pythagorean identity, sin²θ + cos²θ = 1, is fundamental in trigonometry. It allows substitution and simplification of expressions involving sine and cosine powers, which is essential when verifying or transforming trigonometric equations.