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

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Recall the definitions of the trigonometric functions in terms of sine and cosine: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) and \(\sec \theta = \frac{1}{\cos \theta}\).

Rewrite the given expression \(\frac{\tan(-\theta)}{\sec \theta}\) using these definitions: \(\frac{\frac{\sin(-\theta)}{\cos(-\theta)}}{\frac{1}{\cos \theta}}\).

Use the even-odd properties of sine and cosine: \(\sin(-\theta) = -\sin \theta\) (odd function) and \(\cos(-\theta) = \cos \theta\) (even function), so substitute these into the expression.

Simplify the complex fraction by multiplying the numerator by the reciprocal of the denominator: \(\frac{-\sin \theta}{\cos \theta} \times \cos \theta\).

Cancel common factors and write the simplified expression in terms of sine and cosine only, ensuring no quotients remain.

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주요 개념

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Trigonometric Function Definitions

Understanding the basic definitions of trigonometric functions is essential. Tangent (tan θ) is defined as sine over cosine (sin θ / cos θ), and secant (sec θ) is the reciprocal of cosine (1 / cos θ). Expressing all functions in terms of sine and cosine allows for easier manipulation and simplification.

Even-Odd Identities

Even-odd identities describe how trigonometric functions behave with negative angles. For example, sine is an odd function (sin(-θ) = -sin θ), cosine is even (cos(-θ) = cos θ), and tangent is odd (tan(-θ) = -tan θ). Applying these identities helps simplify expressions involving negative angles.

Algebraic Simplification of Trigonometric Expressions

After rewriting functions in terms of sine and cosine, algebraic techniques such as multiplying numerator and denominator, canceling common factors, and eliminating quotients are used to simplify the expression. The goal is to express the result without fractions and only in terms of sine and cosine of θ.

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