Verify that each equation is an identity.
(1 - cos θ)/(1 + cos θ) = 2 csc² θ - 2 csc θ cot θ - 1

Verified step by step guidance

Start by expressing all trigonometric functions in terms of sine and cosine. Note that \( \csc \theta = \frac{1}{\sin \theta} \) and \( \cot \theta = \frac{\cos \theta}{\sin \theta} \).

Rewrite the right-hand side of the equation: \( 2 \csc^2 \theta - 2 \csc \theta \cot \theta - 1 \) becomes \( 2 \left(\frac{1}{\sin^2 \theta}\right) - 2 \left(\frac{1}{\sin \theta}\right)\left(\frac{\cos \theta}{\sin \theta}\right) - 1 \).

Simplify the expression: \( 2 \left(\frac{1}{\sin^2 \theta}\right) - 2 \left(\frac{\cos \theta}{\sin^2 \theta}\right) - 1 \) becomes \( \frac{2 - 2\cos \theta - \sin^2 \theta}{\sin^2 \theta} \).

Use the Pythagorean identity \( \sin^2 \theta = 1 - \cos^2 \theta \) to simplify further: \( \frac{2 - 2\cos \theta - (1 - \cos^2 \theta)}{\sin^2 \theta} \) simplifies to \( \frac{1 - \cos \theta}{1 + \cos \theta} \).

Compare the simplified right-hand side with the left-hand side \( \frac{1 - \cos \theta}{1 + \cos \theta} \) to verify that both sides are equal, confirming the identity.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 co-function identities. Understanding these identities is crucial for simplifying trigonometric expressions and verifying equations.

Cosecant and Cotangent Functions

Cosecant (csc) and cotangent (cot) are two of the six fundamental trigonometric functions. Cosecant is the reciprocal of sine (csc θ = 1/sin θ), while cotangent is the reciprocal of tangent (cot θ = cos θ/sin θ). Familiarity with these functions is essential for manipulating and transforming trigonometric equations.

Algebraic Manipulation in Trigonometry

Algebraic manipulation involves rearranging and simplifying expressions using algebraic techniques. In trigonometry, this includes factoring, combining like terms, and applying identities to transform one side of an equation to match the other. Mastery of these skills is necessary for verifying trigonometric identities effectively.

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Textbook Question

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.

csc θ - sin θ

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Textbook Question

Identify the basic trigonometric function graphed, and determine whether it is even or odd.

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Textbook Question

Find the remaining five trigonometric functions of θ.

cos θ = -1/4, sin θ > 0

Textbook Question

Verify that each equation is an identity.

tan² α sin² α = tan² α + cos² α - 1

Textbook Question