In Exercises 33–42, let sin t = a, cos t = b, and tan t = c. Write each expression in terms of a, b, and c. tan(-t) - tan t

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Recall the identity for the tangent of a negative angle: \(\tan(-t) = -\tan t\).

Substitute the given value \(\tan t = c\) into the expression: \(\tan(-t) - \tan t = -c - c\).

Combine like terms: \(-c - c = -2c\).

Express the final result in terms of \(a\), \(b\), and \(c\). Since the expression only involves \(c\), the answer is \(-2c\).

Thus, the expression \(\tan(-t) - \tan t\) simplifies to \(-2c\) using the given definitions.

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Trigonometric Identities for Negative Angles

Understanding how trigonometric functions behave with negative angles is essential. For tangent, tan(-t) = -tan t, reflecting the odd function property. This identity helps simplify expressions involving negative angles.

Definition of Tangent in Terms of Sine and Cosine

Tangent is defined as the ratio of sine to cosine: tan t = sin t / cos t. Knowing this relationship allows rewriting tangent expressions using sine and cosine values, which is useful when expressing results in terms of a and b.

Algebraic Manipulation of Trigonometric Expressions

Simplifying expressions like tan(-t) - tan t requires algebraic skills to combine and reduce terms. Recognizing patterns and substituting known values (a, b, c) enables expressing the result clearly and concisely.