Verify that each equation is an identity.
sin θ/(1 - cos θ) - sin θ cos θ/( 1 + cos θ) = csc θ (1 + cos² θ)

Verified step by step guidance

Step 1: Start by simplifying the left-hand side (LHS) of the equation: \( \frac{\sin \theta}{1 - \cos \theta} - \frac{\sin \theta \cos \theta}{1 + \cos \theta} \).

Step 2: Find a common denominator for the fractions on the LHS: \((1 - \cos \theta)(1 + \cos \theta) = 1 - \cos^2 \theta = \sin^2 \theta\).

Step 3: Rewrite each fraction with the common denominator: \( \frac{\sin \theta (1 + \cos \theta) - \sin \theta \cos \theta (1 - \cos \theta)}{\sin^2 \theta} \).

Step 4: Simplify the numerator: \( \sin \theta + \sin \theta \cos \theta - \sin \theta \cos \theta + \sin \theta \cos^2 \theta = \sin \theta + \sin \theta \cos^2 \theta \).

Step 5: Factor out \( \sin \theta \) from the numerator: \( \frac{\sin \theta (1 + \cos^2 \theta)}{\sin^2 \theta} = \csc \theta (1 + \cos^2 \theta) \), which matches the right-hand side (RHS).

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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 as identities.

Reciprocal Functions

Reciprocal functions in trigonometry relate the sine, cosine, and tangent functions to their respective cosecant, secant, and cotangent functions. For example, csc θ is the reciprocal of sin θ, defined as 1/sin θ. Recognizing these relationships helps in transforming and simplifying expressions, particularly when verifying identities.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions using algebraic rules. This includes factoring, combining like terms, and applying common denominators. Mastery of these techniques is essential for verifying trigonometric identities, as it allows one to transform one side of the equation to match the other.

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