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

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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.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.