Use identities to write each expression in terms of sin θ and cos θ, and then simplify so that no quotients appear and all functions are of θ only.
cot(-θ)/sec(-θ)

Verified step by step guidance

1. Recognize that \( \cot(-\theta) = \frac{\cos(-\theta)}{\sin(-\theta)} \) and \( \sec(-\theta) = \frac{1}{\cos(-\theta)} \).
2. Use the even-odd identities: \( \cos(-\theta) = \cos(\theta) \) and \( \sin(-\theta) = -\sin(\theta) \).
3. Substitute these identities into the expression: \( \frac{\cos(\theta)}{-\sin(\theta)} \div \frac{1}{\cos(\theta)} \).
4. Simplify the division by multiplying by the reciprocal: \( \frac{\cos(\theta)}{-\sin(\theta)} \times \cos(\theta) \).
5. Combine and simplify the expression: \( \frac{\cos^2(\theta)}{-\sin(\theta)} \).

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 identities, reciprocal identities, and co-function identities. Understanding these identities is essential for rewriting trigonometric expressions in terms of sine and cosine, as they provide the necessary relationships between different functions.

Even and Odd Functions

In trigonometry, even and odd functions have specific properties that affect their behavior under transformations. The cosine function is even, meaning cos(-θ) = cos(θ), while the sine function is odd, so sin(-θ) = -sin(θ). Recognizing these properties is crucial when simplifying expressions involving negative angles, as they help in determining the signs of the resulting functions.

Simplification of Trigonometric Expressions

Simplification involves rewriting trigonometric expressions to eliminate complex forms, such as quotients, and express them solely in terms of sine and cosine. This process often utilizes identities and properties of trigonometric functions to achieve a more straightforward representation. Mastering simplification techniques is vital for solving trigonometric problems efficiently and accurately.