In Exercises 59–68, verify each identity. $\cot \frac{x}{2}=\frac{\sin x}{1-\cos x}$

Verified step by step guidance

1. Start by writing down the given identity clearly: \( \cot \left( \frac{x}{2} \right) = \frac{\sin x}{1 - \cos x} \). Our goal is to verify this identity, meaning we want to show that both sides are equal for all valid values of \( x \).
2. Recall the definition of cotangent in terms of sine and cosine: \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). Apply this to the left side: \( \cot \left( \frac{x}{2} \right) = \frac{\cos \left( \frac{x}{2} \right)}{\sin \left( \frac{x}{2} \right)} \).
3. Use the half-angle formulas for sine and cosine: \[ \sin x = 2 \sin \left( \frac{x}{2} \right) \cos \left( \frac{x}{2} \right) \quad \text{and} \quad 1 - \cos x = 2 \sin^2 \left( \frac{x}{2} \right) \]. These will help rewrite the right side in terms of \( \sin \left( \frac{x}{2} \right) \) and \( \cos \left( \frac{x}{2} \right) \).
4. Substitute these half-angle expressions into the right side: \[ \frac{\sin x}{1 - \cos x} = \frac{2 \sin \left( \frac{x}{2} \right) \cos \left( \frac{x}{2} \right)}{2 \sin^2 \left( \frac{x}{2} \right)} \]. Simplify the fraction by canceling common factors.
5. After simplification, you should get \( \frac{\cos \left( \frac{x}{2} \right)}{\sin \left( \frac{x}{2} \right)} \), which matches the left side expression for \( \cot \left( \frac{x}{2} \right) \). This confirms the identity is true.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. Verifying an identity means showing that both sides of the equation simplify to the same expression using known identities like Pythagorean, reciprocal, or quotient identities.

Cotangent and Its Relationship to Sine and Cosine

Cotangent (cot x) is the reciprocal of tangent and can be expressed as cos x divided by sin x. Understanding this relationship helps in rewriting expressions involving cotangent in terms of sine and cosine, which is often necessary for simplifying or verifying identities.

Algebraic Manipulation of Trigonometric Expressions

Simplifying or verifying identities requires skillful algebraic manipulation, including factoring, expanding, and combining fractions. Recognizing common denominators and applying algebraic techniques alongside trigonometric identities is essential to transform one side of the equation to match the other.