Verify that each equation is an identity.
sin³ θ = sin θ - cos² θ sin θ

Verified step by step guidance

1. Start by examining the right-hand side (RHS) of the equation: \(\sin \theta - \cos^{2} \theta \sin \theta\).
2. Factor out \(\sin \theta\) from the RHS to simplify the expression: \(\sin \theta (1 - \cos^{2} \theta)\).
3. Recall the Pythagorean identity: \(\sin^{2} \theta + \cos^{2} \theta = 1\), which implies \(1 - \cos^{2} \theta = \sin^{2} \theta\).
4. Substitute \(1 - \cos^{2} \theta\) with \(\sin^{2} \theta\) in the factored expression: \(\sin \theta \cdot \sin^{2} \theta\).
5. Simplify the expression to get \(\sin^{3} \theta\), which matches the left-hand side (LHS), thus verifying the 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 identities like Pythagorean or angle sum formulas.

Pythagorean Identity

The Pythagorean identity states that sin²θ + cos²θ = 1 for any angle θ. This fundamental relationship allows substitution between sine and cosine terms, which is essential for simplifying and verifying trigonometric expressions.

Algebraic Manipulation of Trigonometric Expressions

Simplifying trigonometric expressions often requires factoring, expanding, or rearranging terms. Recognizing patterns like factoring out common terms (e.g., sin θ) helps transform one side of the equation to match the other, confirming the identity.