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.
cos θ (cos θ - sec θ)

검증된 단계별 안내

Start by rewriting the given expression \(\cos \theta (\cos \theta - \sec \theta)\), focusing on expressing all trigonometric functions in terms of sine and cosine.

Recall that \(\sec \theta\) is the reciprocal of \(\cos \theta\), so rewrite \(\sec \theta\) as \(\frac{1}{\cos \theta}\).

Substitute \(\sec \theta\) in the expression to get \(\cos \theta \left( \cos \theta - \frac{1}{\cos \theta} \right)\).

Distribute \(\cos \theta\) across the terms inside the parentheses: \(\cos \theta \cdot \cos \theta - \cos \theta \cdot \frac{1}{\cos \theta}\).

Simplify each term: \(\cos^2 \theta - 1\), ensuring no quotients remain and all functions are in terms of \(\theta\) only.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

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주요 개념

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

Trigonometric Functions and Their Reciprocal Identities

Understanding the basic trigonometric functions sine and cosine, along with their reciprocal functions such as secant (sec θ = 1/cos θ), is essential. This allows rewriting expressions involving secant in terms of cosine, facilitating simplification.

Expression Simplification Without Quotients

Simplifying trigonometric expressions to eliminate quotients involves rewriting all terms so that no fractions remain. This often requires multiplying through by denominators or substituting reciprocal identities to express everything in sine and cosine only.

Algebraic Manipulation of Trigonometric Expressions

Skillful algebraic manipulation, such as distributing terms, combining like terms, and factoring, is necessary to simplify trigonometric expressions. This helps in rewriting the expression in a cleaner form involving only sine and cosine functions of θ.

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.cos θ (cos θ - sec θ)