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

1. Start by expressing all trigonometric functions in terms of sine and cosine. Note that \( \csc \theta = \frac{1}{\sin \theta} \) and \( \cot \theta = \frac{\cos \theta}{\sin \theta} \).
2. Rewrite the right-hand side of the equation: \( 2 \csc^2 \theta - 2 \csc \theta \cot \theta - 1 \) becomes \( 2 \left(\frac{1}{\sin^2 \theta}\right) - 2 \left(\frac{1}{\sin \theta}\right)\left(\frac{\cos \theta}{\sin \theta}\right) - 1 \).
3. Simplify the expression: \( 2 \left(\frac{1}{\sin^2 \theta}\right) - 2 \left(\frac{\cos \theta}{\sin^2 \theta}\right) - 1 \) becomes \( \frac{2 - 2\cos \theta - \sin^2 \theta}{\sin^2 \theta} \).
4. 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 \( \frac{1 - \cos \theta}{1 + \cos \theta} \).
5. 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.

Key Concepts

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.