In Exercises 1–60, verify each identity. tan (-x) cos x = -sin x

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Start by recalling the trigonometric identity for the tangent of a negative angle: \( \tan(-x) = -\tan(x) \).

Substitute \( \tan(-x) \) with \( -\tan(x) \) in the given expression: \( \tan(-x) \cos(x) = -\tan(x) \cos(x) \).

Use the definition of tangent in terms of sine and cosine: \( \tan(x) = \frac{\sin(x)}{\cos(x)} \).

Substitute \( \tan(x) \) with \( \frac{\sin(x)}{\cos(x)} \) in the expression: \( -\tan(x) \cos(x) = -\left(\frac{\sin(x)}{\cos(x)}\right) \cos(x) \).

Simplify the expression by canceling \( \cos(x) \) in the numerator and denominator: \( -\sin(x) \). This shows that \( \tan(-x) \cos(x) = -\sin(x) \), verifying the identity.

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Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. Common identities include the Pythagorean identities, reciprocal identities, and co-function identities. Understanding these identities is crucial for simplifying expressions and verifying equations in trigonometry.

Tangent Function and Its Properties

The tangent function, defined as the ratio of the sine and cosine functions (tan(x) = sin(x)/cos(x)), has specific properties, including periodicity and symmetry. Notably, tan(-x) = -tan(x), which reflects the odd nature of the tangent function. This property is essential for manipulating and verifying trigonometric identities.

Negative Angle Identities

Negative angle identities express the values of trigonometric functions for negative angles. For example, sin(-x) = -sin(x) and cos(-x) = cos(x). These identities help in transforming expressions involving negative angles into more manageable forms, which is particularly useful in verifying identities like the one presented in the question.