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

Verified step by step guidance

1. Start by expressing \( \cot \theta \) and \( \tan \theta \) in terms of \( \sin \theta \) and \( \cos \theta \). Recall that \( \cot \theta = \frac{\cos \theta}{\sin \theta} \) and \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).
2. Rewrite the right-hand side of the equation using these identities: \( \frac{\sin \theta}{1 - \frac{\cos \theta}{\sin \theta}} + \frac{\cos \theta}{1 - \frac{\sin \theta}{\cos \theta}} \).
3. Simplify each fraction by finding a common denominator for the terms in the denominators: \( \frac{\sin \theta}{\frac{\sin \theta - \cos \theta}{\sin \theta}} + \frac{\cos \theta}{\frac{\cos \theta - \sin \theta}{\cos \theta}} \).
4. Simplify further by multiplying the numerators by the reciprocal of the denominators: \( \frac{\sin^2 \theta}{\sin \theta - \cos \theta} + \frac{\cos^2 \theta}{\cos \theta - \sin \theta} \).
5. Combine the fractions over a common denominator and simplify the expression to verify if it equals \( \sin \theta + \cos \theta \).

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 quotient identities. Understanding these identities is crucial for simplifying trigonometric expressions and verifying equations as identities.

Cotangent and Tangent Functions

The cotangent (cot) and tangent (tan) functions are fundamental trigonometric functions defined as cot θ = cos θ/sin θ and tan θ = sin θ/cos θ, respectively. These functions are essential for manipulating and transforming trigonometric expressions, especially when verifying identities that involve these ratios.

Common Denominators

Finding a common denominator is a key technique in algebra that allows for the addition or comparison of fractions. In the context of trigonometric identities, it is often necessary to express terms with different denominators in a unified form to facilitate simplification and verification of the identity. This concept is particularly relevant when dealing with expressions involving cotangent and tangent.