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. Rewrite \( \sec \theta \) in terms of \( \cos \theta \): \( \sec \theta = \frac{1}{\cos \theta} \).
2. Substitute \( \sec \theta \) in the expression: \( \cos \theta (\cos \theta - \frac{1}{\cos \theta}) \).
3. Distribute \( \cos \theta \) across the terms inside the parentheses: \( \cos \theta \cdot \cos \theta - \cos \theta \cdot \frac{1}{\cos \theta} \).
4. Simplify each term: \( \cos^2 \theta - 1 \).
5. Recognize that \( \cos^2 \theta - 1 \) can be rewritten using the Pythagorean identity: \( \cos^2 \theta - 1 = -\sin^2 \theta \).

Key Concepts

Trigonometric Functions

Trigonometric functions, such as sine (sin) and cosine (cos), relate the angles of a triangle to the ratios of its sides. Understanding these functions is essential for manipulating and simplifying trigonometric expressions. The secant function (sec) is the reciprocal of cosine, defined as sec θ = 1/cos θ, which is crucial for rewriting expressions in terms of sine and cosine.

Reciprocal Identities

Reciprocal identities are fundamental relationships in trigonometry that express one trigonometric function in terms of another. For example, sec θ = 1/cos θ and csc θ = 1/sin θ. These identities allow us to convert expressions involving secant into forms that only involve sine and cosine, facilitating simplification and manipulation of trigonometric expressions.

Simplification of Trigonometric Expressions

Simplifying trigonometric expressions involves rewriting them to eliminate quotients and express them solely in terms of sine and cosine. This process often includes applying identities, factoring, and combining like terms. The goal is to achieve a more manageable form that can be easily analyzed or solved, which is particularly important in solving trigonometric equations.