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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 5.62
Capitolo 6, Problema 5.62

Verify that each equation is an identity.
1/(sec α - tan α) = sec α + tan α

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Start by expressing \( \sec \alpha \) and \( \tan \alpha \) in terms of sine and cosine: \( \sec \alpha = \frac{1}{\cos \alpha} \) and \( \tan \alpha = \frac{\sin \alpha}{\cos \alpha} \).
Substitute these expressions into the left-hand side of the equation: \( \frac{1}{\sec \alpha - \tan \alpha} = \frac{1}{\frac{1}{\cos \alpha} - \frac{\sin \alpha}{\cos \alpha}} \).
Simplify the denominator by combining the fractions: \( \frac{1}{\frac{1 - \sin \alpha}{\cos \alpha}} = \frac{\cos \alpha}{1 - \sin \alpha} \).
Now, consider the right-hand side: \( \sec \alpha + \tan \alpha = \frac{1}{\cos \alpha} + \frac{\sin \alpha}{\cos \alpha} = \frac{1 + \sin \alpha}{\cos \alpha} \).
Multiply the numerator and the denominator of the left-hand side by the conjugate of the denominator \( 1 + \sin \alpha \) to simplify and verify that both sides are equal: \( \frac{\cos \alpha (1 + \sin \alpha)}{(1 - \sin \alpha)(1 + \sin \alpha)} = \frac{1 + \sin \alpha}{\cos \alpha} \).

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Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the variable where both sides are defined. Common identities include the Pythagorean identities, reciprocal identities, and quotient identities. Understanding these identities is crucial for verifying equations and simplifying expressions in trigonometry.
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Fundamental Trigonometric Identities

Reciprocal Functions

Reciprocal functions in trigonometry include secant (sec) and cosecant (csc), which are the reciprocals of cosine and sine, respectively. The secant function is defined as sec α = 1/cos α, and the tangent function is defined as tan α = sin α/cos α. Recognizing these relationships is essential for manipulating and verifying trigonometric equations.
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Secant, Cosecant, & Cotangent on the Unit Circle

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions using algebraic rules. This includes factoring, expanding, and combining like terms. In the context of trigonometric identities, effective algebraic manipulation allows one to transform one side of an equation to match the other, thereby verifying the identity.
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Algebraic Operations on Vectors
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Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.

csc θ cos θ tan θ

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Identify the basic trigonometric function graphed, and determine whether it is even or odd.


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Find the remaining five trigonometric functions of θ.

cos θ = 1/5, θ in quadrant I

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Verify that each equation is an identity.

(1 + sin θ)/(1 - sin θ) - (1 - sin θ)/( 1 + sin θ) = 4 tan θ sec θ

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Verify that each equation is an identity.

sec⁴ x - sec² x = tan⁴ x + tan² x

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Verify that each equation is an identity.

sin θ/(1 - cos θ) - sin θ cos θ/( 1 + cos θ) = csc θ (1 + cos² θ)

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