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Ch. 3 - Trigonometric Identities and Equations
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 3rd EditionCh. 3 - Trigonometric Identities and EquationsProblema 11
Capitolo 3, Problema 11

Verify each identity. csc θ - sin θ = cot θ cos θ

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Start by writing down the given identity to verify: \(\csc \theta - \sin \theta = \cot \theta \cos \theta\).
Recall the fundamental trigonometric definitions: \(\csc \theta = \frac{1}{\sin \theta}\) and \(\cot \theta = \frac{\cos \theta}{\sin \theta}\).
Rewrite the left-hand side (LHS) using the definition of cosecant: \(\csc \theta - \sin \theta = \frac{1}{\sin \theta} - \sin \theta\).
Find a common denominator on the LHS to combine the terms: \(\frac{1}{\sin \theta} - \sin \theta = \frac{1 - \sin^2 \theta}{\sin \theta}\).
Use the Pythagorean identity \(1 - \sin^2 \theta = \cos^2 \theta\) to simplify the numerator, so the LHS becomes \(\frac{\cos^2 \theta}{\sin \theta}\). Then compare this with the right-hand side (RHS) \(\cot \theta \cos \theta = \frac{\cos \theta}{\sin \theta} \times \cos \theta = \frac{\cos^2 \theta}{\sin \theta}\) to verify the identity.

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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 of the equation are equivalent by using known identities and algebraic manipulation.
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Fundamental Trigonometric Identities

Reciprocal and Quotient Identities

Reciprocal identities relate sine, cosine, and tangent to their reciprocal functions: cosecant (csc), secant (sec), and cotangent (cot). For example, csc θ = 1/sin θ and cot θ = cos θ/sin θ. These are essential for rewriting expressions to verify identities.
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Algebraic Manipulation of Trigonometric Expressions

Algebraic manipulation involves factoring, combining fractions, and simplifying expressions using trigonometric identities. This skill is crucial to transform one side of the identity into the other, often by expressing all terms in sine and cosine.
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