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Ch. 5 - Trigonometric Identities
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoLial 12th EditionCh. 5 - Trigonometric IdentitiesProblema 68
Capitolo 6, Problema 68

Verify that each equation is an identity.
(1/2)cot (x/2) - (1/2) tan (x/2) = cot x

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Start by recalling the definitions of cotangent and tangent in terms of sine and cosine: \(\cot \theta = \frac{\cos \theta}{\sin \theta}\) and \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).
Rewrite the left-hand side (LHS) of the equation \(\frac{1}{2} \cot \left( \frac{x}{2} \right) - \frac{1}{2} \tan \left( \frac{x}{2} \right)\) using these definitions: express both cotangent and tangent as fractions involving sine and cosine.
Combine the two terms on the LHS over a common denominator to simplify the expression. This will involve algebraic manipulation of fractions.
Use the double-angle identities for sine and cosine, specifically \(\sin x = 2 \sin \frac{x}{2} \cos \frac{x}{2}\) and \(\cos x = \cos^2 \frac{x}{2} - \sin^2 \frac{x}{2}\), to rewrite the expression in terms of \(\sin x\) and \(\cos x\).
After simplification, compare the resulting expression to the right-hand side (RHS) \(\cot x = \frac{\cos x}{\sin x}\). If both sides match, the identity is verified.

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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 manipulating one side of the equation using known identities until it matches the other side, confirming their equivalence.
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Fundamental Trigonometric Identities

Half-Angle Formulas

Half-angle formulas express trigonometric functions of half an angle in terms of the full angle. For example, cot(x/2) and tan(x/2) can be related to functions of x, which helps simplify or transform expressions involving half angles.
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Quadratic Formula

Cotangent and Tangent Relationship

Cotangent and tangent are reciprocal functions: cot(θ) = 1/tan(θ). Understanding their relationship allows for rewriting expressions involving cot and tan, facilitating simplification and verification of identities.
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Introduction to Cotangent Graph
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