In Exercises 1–60, verify each identity.cos θ sec θ ----------------- = tan θcot θ

Verified step by step guidance

1. Start by recalling the definitions of the trigonometric functions involved: \( \sec \theta = \frac{1}{\cos \theta} \), \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), and \( \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} \).
2. Substitute \( \sec \theta = \frac{1}{\cos \theta} \) into the left side of the equation: \( \frac{\cos \theta \cdot \frac{1}{\cos \theta}}{\cot \theta} \).
3. Simplify the numerator: \( \cos \theta \cdot \frac{1}{\cos \theta} = 1 \).
4. Now the expression becomes \( \frac{1}{\cot \theta} \).
5. Recognize that \( \frac{1}{\cot \theta} = \tan \theta \), thus verifying the identity.

Key Concepts

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 quotient identities. Understanding these identities is crucial for verifying equations and simplifying expressions in trigonometry.

Reciprocal Functions

Reciprocal functions in trigonometry refer to pairs of functions where one function is the reciprocal of another. For example, the secant function (sec θ) is the reciprocal of the cosine function (cos θ), and the cosecant function (csc θ) is the reciprocal of the sine function (sin θ). Recognizing these relationships is essential for manipulating and verifying trigonometric identities.

Quotient Identity

The quotient identity in trigonometry states that the tangent function (tan θ) is equal to the sine function (sin θ) divided by the cosine function (cos θ). This identity is fundamental for transforming and simplifying expressions involving tangent, cotangent, and other trigonometric functions. It plays a key role in verifying identities like the one presented in the question.