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.
tan(-θ)/sec θ

Verified step by step guidance

1. Recall that \( \tan(-\theta) = -\tan(\theta) \) and \( \sec(\theta) = \frac{1}{\cos(\theta)} \).
2. Express \( \tan(\theta) \) in terms of sine and cosine: \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \).
3. Substitute these into the expression: \( \frac{\tan(-\theta)}{\sec(\theta)} = \frac{-\tan(\theta)}{\sec(\theta)} = \frac{-\frac{\sin(\theta)}{\cos(\theta)}}{\frac{1}{\cos(\theta)}} \).
4. Simplify the expression by multiplying the numerator by the reciprocal of the denominator: \( -\frac{\sin(\theta)}{\cos(\theta)} \times \cos(\theta) \).
5. Cancel out \( \cos(\theta) \) in the numerator and denominator to get \( -\sin(\theta) \).

Key Concepts

Trigonometric Functions

Trigonometric functions, including sine (sin), cosine (cos), and tangent (tan), are fundamental in trigonometry. They relate the angles of a triangle to the ratios of its sides. For example, tan(θ) is defined as sin(θ)/cos(θ). Understanding these functions is essential for manipulating and simplifying trigonometric expressions.

Reciprocal Identities

Reciprocal identities express trigonometric functions in terms of their reciprocals. For instance, sec(θ) is the reciprocal of cos(θ), meaning sec(θ) = 1/cos(θ). These identities are crucial for rewriting expressions and simplifying them, especially when aiming to eliminate quotients in trigonometric expressions.

Even and Odd Functions

In trigonometry, sine and tangent are odd functions, while cosine is an even function. This means that sin(-θ) = -sin(θ) and cos(-θ) = cos(θ). Recognizing these properties helps in simplifying expressions involving negative angles, which is important for the given problem of simplifying tan(-θ)/sec(θ).