Problem 65 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

The given identity is valid.

Step 1: Simplify the Right Hand Side(RHS)

Let’s start by simplifying the RHS of the equation: \(\frac{\sin x}{1-\cos x}\). We can rewrite \(\sin x\) as \(2 \sin \frac{x}{2} \cos \frac{x}{2}\) and \(\cos x\) as \(1 - 2\sin^2 \frac{x}{2}\) using the double angle identities. Thus, RHS becomes: \(\frac{2 \sin \frac{x}{2} \cos \frac{x}{2}}{1- (1 - 2\sin^2 \frac{x}{2})} = \frac{2 \sin \frac{x}{2} \cos \frac{x}{2}}{2\sin^2 \frac{x}{2}}\).

Step 2: Simplify Further

On further simplification of the RHS from step 1, we get: \(\frac{\cos \frac{x}{2}}{\sin \frac{x}{2}}\).

Step 3: Conversion to Cotangent

We know that the cotangent is the reciprocal of the tangent. So, \(\frac{\cos \frac{x}{2}}{\sin \frac{x}{2}}\) can be rewritten as \(\cot \frac{x}{2}\). Therefore, we have shown that the RHS simplifies to the LHS.

Key Concepts

Cotangent

Understanding the concept of cotangent can greatly aid in simplifying and verifying trigonometric identities. The cotangent function, \( \cot \theta \), is the reciprocal of the tangent function. That means:

• \( \cot \theta = \frac{1}{\tan \theta} \)
• It can also be expressed as \( \frac{\cos \theta}{\sin \theta} \)

Cotangent specifically refers to the ratio of the adjacent side to the opposite side in a right-angled triangle. In this context, we used the identity: \( \cot \frac{x}{2} = \frac{\cos \frac{x}{2}}{\sin \frac{x}{2}} \), which simplifies the challenge of proving identities. This is because many trigonometric identities can be boiled down to these basic functions and their relationships, simplifying complex expressions and making them easier to verify or deduce. Knowing these base conversions is critical for mastering trigonometry.

Double Angle Identities

Double angle identities are pivotal in trigonometry as they allow us to express trigonometric functions of double angles in terms of single angles. They are particularly useful in simplifying expressions or solving equations.

For sine and cosine, the double angle identities are:

• \( \sin 2\theta = 2\sin \theta \cos \theta \)
• \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \)
These can also be rewritten in different forms, such as:
• \( \cos 2\theta = 1 - 2\sin^2 \theta \)
• \( \cos 2\theta = 2\cos^2 \theta - 1 \)
In the exercise provided, these identities were applied to rewrite \( \sin x \) and \( \cos x \) in terms of \( \sin \frac{x}{2} \) and \( \cos \frac{x}{2} \).

These transformations reduce the complexity of verifying trigonometric identities, as seen in our original problem of rewriting \( \frac{\sin x}{1-\cos x} \) using double-angle identities before simplifying.

Simplification of Trigonometric Expressions

Simplifying trigonometric expressions is a foundational skill in solving trigonometric identities and equations. The goal is usually to transform expressions into more manageable forms using identities and mathematical operations.

The following steps are often helpful when simplifying trigonometric expressions:

• Use known identities, such as Pythagorean identities, double angle identities, and reciprocal identities like \( \cot \theta = \frac{1}{\tan \theta} \).
• Factor expressions where possible to reveal common patterns and relationships.
• Simplify fractions by finding common denominators or using reciprocal identities.

In our exercise, we simplified \( \frac{\sin x}{1-\cos x} \) by transforming the numerator and the denominator using double angle identities, eventually arriving at \( \cot \frac{x}{2} \).

This process illustrates the strategic manipulation of expressions using a combination of identities and algebraic skills, a crucial aspect for students aiming to master trigonometry.