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.
-sec² (-θ) + sin² (-θ) + cos² (-θ)

Verified step by step guidance

1. Recognize that \( \sec(-\theta) = \frac{1}{\cos(-\theta)} \) and use the even property of cosine: \( \cos(-\theta) = \cos(\theta) \).
2. Rewrite \( \sec^2(-\theta) \) as \( \frac{1}{\cos^2(\theta)} \).
3. Use the identity \( \sin^2(-\theta) = \sin^2(\theta) \) because sine is an odd function.
4. Use the Pythagorean identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \).
5. Substitute and simplify the expression: \( -\frac{1}{\cos^2(\theta)} + \sin^2(\theta) + \cos^2(\theta) \) using the identity to eliminate quotients.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. Key identities include the Pythagorean identity, which states that sin²(θ) + cos²(θ) = 1, and the reciprocal identities, such as sec(θ) = 1/cos(θ). Understanding these identities is essential for simplifying trigonometric expressions.

Even and Odd Functions

In trigonometry, functions are classified as even or odd based on their symmetry. Even functions, like cosine, satisfy the property f(-θ) = f(θ), while odd functions, like sine, satisfy f(-θ) = -f(θ). Recognizing these properties helps in simplifying expressions involving negative angles, as it allows for the substitution of equivalent positive angle expressions.

Simplification of Trigonometric Expressions

Simplifying trigonometric expressions involves rewriting them in a more manageable form, often using identities to eliminate quotients and combine terms. This process may include converting all functions to sine and cosine, as well as applying identities to reduce the expression to its simplest form. Mastery of simplification techniques is crucial for solving complex trigonometric problems.