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)

Verified step by step guidance

1. Start by rewriting the given expression \(\cos \theta (\cos \theta - \sec \theta)\), focusing on expressing all trigonometric functions in terms of sine and cosine.
2. Recall that \(\sec \theta\) is the reciprocal of \(\cos \theta\), so rewrite \(\sec \theta\) as \(\frac{1}{\cos \theta}\).
3. Substitute \(\sec \theta\) in the expression to get \(\cos \theta \left( \cos \theta - \frac{1}{\cos \theta} \right)\).
4. Distribute \(\cos \theta\) across the terms inside the parentheses: \(\cos \theta \cdot \cos \theta - \cos \theta \cdot \frac{1}{\cos \theta}\).
5. Simplify each term: \(\cos^2 \theta - 1\), ensuring no quotients remain and all functions are in terms of \(\theta\) only.

Key Concepts

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.