Ch. 1 - Angles and the Trigonometric Functions

Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook

Blitzer 3rd Edition

Ch. 1 - Angles and the Trigonometric Functions

Problem 29

Chapter 1, Problem 29

In Exercises 25–30, use an identity to find the value of each expression. Do not use a calculator. sec² 23° - tan² 23°

Verified step by step guidance

Recall the Pythagorean identity involving secant and tangent: \(\sec^{2} \theta - \tan^{2} \theta = 1\).

Identify the angle in the problem: here, \(\theta = 23^\circ\).

Apply the identity directly by substituting \(\theta = 23^\circ\) into the expression: \(\sec^{2} 23^\circ - \tan^{2} 23^\circ\).

Since the identity holds for all angles where these functions are defined, the expression simplifies to 1 without further calculation.

Therefore, the value of \(\sec^{2} 23^\circ - \tan^{2} 23^\circ\) is 1 by the Pythagorean identity.

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Key Concepts

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

Pythagorean Identity for Secant and Tangent

The identity sec²θ - tan²θ = 1 is a fundamental Pythagorean identity in trigonometry. It relates the secant and tangent functions of the same angle and allows simplification of expressions without a calculator.

Definition of Secant and Tangent Functions

Secant (sec θ) is the reciprocal of cosine (1/cos θ), and tangent (tan θ) is the ratio of sine to cosine (sin θ/cos θ). Understanding these definitions helps in applying identities and simplifying trigonometric expressions.

Using Identities to Simplify Expressions

Trigonometric identities allow rewriting complex expressions into simpler forms. Recognizing which identity applies enables solving problems efficiently without numerical approximation or calculators.

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