In Exercises 1–60, verify each identity.csc θ - sin θ = cot θ cos θ

Verified step by step guidance

1. Start by expressing all trigonometric functions in terms of sine and cosine. Recall that \( \csc \theta = \frac{1}{\sin \theta} \) and \( \cot \theta = \frac{\cos \theta}{\sin \theta} \).
2. Rewrite the left side of the equation: \( \csc \theta - \sin \theta = \frac{1}{\sin \theta} - \sin \theta \).
3. Combine the terms on the left side over a common denominator: \( \frac{1}{\sin \theta} - \sin \theta = \frac{1 - \sin^2 \theta}{\sin \theta} \).
4. Use the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \) to replace \( 1 - \sin^2 \theta \) with \( \cos^2 \theta \).
5. Now, the expression becomes \( \frac{\cos^2 \theta}{\sin \theta} \), which can be rewritten as \( \frac{\cos \theta}{\sin \theta} \cdot \cos \theta = \cot \theta \cos \theta \), matching the right side of the identity.

Key Concepts

Cosecant and Sine Functions

The cosecant function, denoted as csc θ, is the reciprocal of the sine function. This means csc θ = 1/sin θ. Understanding the relationship between these two functions is crucial for manipulating trigonometric identities, as it allows for the conversion between different forms of expressions involving sine.

Cotangent and Cosine Functions

The cotangent function, represented as cot θ, is the ratio of the cosine function to the sine function, defined as cot θ = cos θ/sin θ. This relationship is essential for verifying trigonometric identities, as it enables the substitution of cotangent in terms of sine and cosine, facilitating simplification of expressions.

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the variable where both sides are defined. Common identities include the Pythagorean identities, reciprocal identities, and quotient identities. Familiarity with these identities is vital for verifying equations, as they provide the foundational tools needed to manipulate and simplify trigonometric expressions.