Problem 12 - Algebra 2

Question

Verify that each of the following is an identity. $$ \frac{1-\cos \theta}{1+\cos \theta}=(\csc \theta-\cot \theta)^{2} $$

Step-by-Step Solution

Answer

The given expression is an identity because both sides simplify to the same form.

Step 1: Rewrite the Left Side

Rewrite \( \frac{1 - \cos \theta}{1 + \cos \theta} \) using trigonometric identities. We'll try to make it look more like the identity on the right side. Remember that \(1 - \cos^2 \theta = \sin^2 \theta \), which might be useful later.

Step 2: Simplify the Right Side

Express \( (\csc \theta - \cot \theta)^{2} \) in terms of sine and cosine. We know that \( \csc \theta = \frac{1}{\sin \theta} \) and \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). This gives:\[(\csc \theta - \cot \theta)^{2} = \left(\frac{1}{\sin \theta} - \frac{\cos \theta}{\sin \theta}\right)^{2}\]\[= \left(\frac{1 - \cos \theta}{\sin \theta}\right)^{2}\]\[= \frac{(1 - \cos \theta)^{2}}{\sin^2 \theta}\].

Step 3: Test Equality

We have two expressions: \( \frac{1 - \cos \theta}{1 + \cos \theta} \) and \( \frac{(1 - \cos \theta)^{2}}{\sin^2 \theta} \). To check identity, multiply and divide the left-hand side by \( (1 - \cos \theta) \) to match the form of the right-hand side:\[\frac{1 - \cos \theta}{1 + \cos \theta} \cdot \frac{1 - \cos \theta}{1 - \cos \theta} = \frac{(1 - \cos \theta)^{2}}{1 - \cos^2 \theta}\]Then remember: \( 1 - \cos^2 \theta = \sin^2 \theta \), so:\[= \frac{(1 - \cos \theta)^{2}}{\sin^2 \theta} \].

Step 4: Verification

Since both sides equal the same expression, we have verified that the given expression is indeed an identity.

Key Concepts

Trigonometry: Understanding the Basics

Trigonometry revolves around the study of angles and the relationships between the angles and sides of triangles. It is one of the critical branches of mathematics that finds applications in various fields such as physics, engineering, and astronomy.

In this exercise, we focus on two specific trigonometric functions: cosecant (\(\csc \theta\)) and cotangent (\(\cot \theta\)).

• Cosecant is the reciprocal of the sine function and is expressed as\(\csc \theta = \frac{1}{\sin \theta} \).


• Cotangent is the ratio of the cosine function to the sine function, expressed as \(\cot \theta = \frac{\cos \theta}{\sin \theta} \)

Understanding these definitions is crucial for manipulating and simplifying trigonometric expressions, which is what we do throughout this problem. Remembering these fundamental relationships will help you to see how the different trigonometric terms relate to one another. This will ultimately ease the process of verifying identities later on.

Algebraic Manipulation in Trigonometry

Algebraic manipulation is a powerful tool in mathematics that allows us to transform and simplify expressions. In trigonometry, this often involves using known identities to rewrite expressions in a different form.

A common identity used is \(1 - \cos^2 \theta = \sin^2 \theta\), which stems from the Pythagorean identity\( \sin^2 \theta + \cos^2 \theta = 1 \).

In this problem, we aim to transform the given expressions to reveal their equality.

• The left side \(\frac{1 - \cos \theta}{1 + \cos \theta}\) is rewritten using the identity above by multiplying and dividing by \(1 - \cos \theta\)


• The right side \((\csc \theta - \cot \theta)^2\)is rewritten by substituting \(\csc \theta\) and \(\cot \theta\) into their sine and cosine forms, giving us a new expression: \(\frac{(1 - \cos \theta)^2}{\sin^2 \theta}\).

By manipulating both sides algebraically, we simplify each to a common form that verifies our identity. This technique is invaluable when working on problems that require proof of equivalent expressions.

Verifying Identities: Proving Equality

Verifying identities is the process of proving that two expressions are equivalent for all values within the functions' domains. It involves a series of transformations, often leveraging trigonometric and algebraic identities.

• In our exercise, we're not just simplifying; we are showing that \(\frac{1 - \cos \theta}{1 + \cos \theta}\) is the same as \((\csc \theta - \cot \theta)^2\).


• The steps involve manipulating each side independently, then testing their equality.

To do this, ensure that each transformation retains the integrity of the expression, maintaining equivalence. We checked that both expressions eventually simplified to \(\frac{(1 - \cos \theta)^2}{\sin^2 \theta}\), thereby proving the identity as true.

Remember, verifying identities can often seem complex, but breaking them down into smaller, manageable steps, as shown here, will simplify the process. Practice and familiarity with common identities will make these problems much more intuitive over time.