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.
(sec θ - 1) (sec θ + 1)

검증된 단계별 안내

Recall the definition of secant in terms of cosine: \(\sec \theta = \frac{1}{\cos \theta}\).

Rewrite the expression \((\sec \theta - 1)(\sec \theta + 1)\) by substituting \(\sec \theta\) with \(\frac{1}{\cos \theta}\), so it becomes \(\left(\frac{1}{\cos \theta} - 1\right) \left(\frac{1}{\cos \theta} + 1\right)\).

Recognize that the expression is a product of conjugates, which follows the difference of squares formula: \((a - b)(a + b) = a^2 - b^2\). Here, \(a = \frac{1}{\cos \theta}\) and \(b = 1\).

Apply the difference of squares formula to get \(\left(\frac{1}{\cos \theta}\right)^2 - 1^2 = \frac{1}{\cos^2 \theta} - 1\).

Rewrite the expression \(\frac{1}{\cos^2 \theta} - 1\) as a single fraction with denominator \(\cos^2 \theta\), resulting in \(\frac{1 - \cos^2 \theta}{\cos^2 \theta}\), and then use the Pythagorean identity \(\sin^2 \theta = 1 - \cos^2 \theta\) to express the numerator in terms of sine.

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주요 개념

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

Definition of Secant in Terms of Cosine

Secant (sec θ) is the reciprocal of cosine, defined as sec θ = 1/cos θ. Expressing secant in terms of cosine allows rewriting expressions involving secant into sine and cosine, which are the fundamental trigonometric functions.

Algebraic Simplification of Trigonometric Expressions

Simplifying trigonometric expressions involves combining like terms, factoring, and eliminating quotients by multiplying numerator and denominator appropriately. This process helps rewrite expressions without fractions and in terms of sine and cosine only.

Pythagorean Identity

The Pythagorean identity states that sin²θ + cos²θ = 1. This identity is essential for simplifying expressions by replacing sin²θ or cos²θ terms, enabling the expression to be written in a simpler form involving only sine and cosine.

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.(sec θ - 1) (sec θ + 1)