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Ch. 5 - Trigonometric Identities
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook
All textbooksLial 12th EditionCh. 5 - Trigonometric IdentitiesProblem 5.1.76
Chapter 6, Problem 5.1.76

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.
tan(-θ)/sec θ

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Recall the definitions of the trigonometric functions in terms of sine and cosine: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) and \(\sec \theta = \frac{1}{\cos \theta}\).
Rewrite the given expression \(\frac{\tan(-\theta)}{\sec \theta}\) using these definitions: \(\frac{\frac{\sin(-\theta)}{\cos(-\theta)}}{\frac{1}{\cos \theta}}\).
Use the even-odd properties of sine and cosine: \(\sin(-\theta) = -\sin \theta\) (odd function) and \(\cos(-\theta) = \cos \theta\) (even function), so substitute these into the expression.
Simplify the complex fraction by multiplying the numerator by the reciprocal of the denominator: \(\frac{-\sin \theta}{\cos \theta} \times \cos \theta\).
Cancel common factors and write the simplified expression in terms of sine and cosine only, ensuring no quotients remain.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Trigonometric Function Definitions

Understanding the basic definitions of trigonometric functions is essential. Tangent (tan θ) is defined as sine over cosine (sin θ / cos θ), and secant (sec θ) is the reciprocal of cosine (1 / cos θ). Expressing all functions in terms of sine and cosine allows for easier manipulation and simplification.
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Introduction to Trigonometric Functions

Even-Odd Identities

Even-odd identities describe how trigonometric functions behave with negative angles. For example, sine is an odd function (sin(-θ) = -sin θ), cosine is even (cos(-θ) = cos θ), and tangent is odd (tan(-θ) = -tan θ). Applying these identities helps simplify expressions involving negative angles.
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Even and Odd Identities

Algebraic Simplification of Trigonometric Expressions

After rewriting functions in terms of sine and cosine, algebraic techniques such as multiplying numerator and denominator, canceling common factors, and eliminating quotients are used to simplify the expression. The goal is to express the result without fractions and only in terms of sine and cosine of θ.
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Related Practice
Textbook Question

Determine whether the positive or negative square root should be selected.

sin (-10°) = ± √[(1 - cos (-20°))/2]

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Textbook Question

Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression.

csc² t - 1

Textbook Question

For each expression in Column I, choose the expression from Column II that completes an identity. One or both expressions may need to be rewritten.

sec x/csc x


II

A. sin ^2 x/cos ^2 x

B.1/(sec ^2 x)

C. sin (-x)

D. csc ^2 x-cot ^2 x + sin ^2 x

E. tan x

Textbook Question

Verify that each equation is an identity.

sin² β (1 + cot² β) = 1

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Textbook Question

Perform each indicated operation and simplify the result so that there are no quotients.

1/( sin α - 1) - 1/(sin α + 1)

Textbook Question

Find the remaining five trigonometric functions of θ.

sin θ = -4/5, cos θ < 0