Problem 50 - Precalculus: Functions and Graphs

Question

Verify the Identity. $$\ln |\csc x-\cot x|=-\ln |\csc x+\cot x|$$

Step-by-Step Solution

Answer

The identity is verified by showing the logarithms sum to zero, confirming they are negatives.

Step 1: Understand the Problem

We need to verify that the given identity holds true. The identity states that \( \ln |\csc x - \cot x| = -\ln |\csc x + \cot x| \). This means that the logarithms of two expressions involving cosecant and cotangent are negatives of each other.

Step 2: Simplify the Expression

Let's start by simplifying \( \ln |\csc x - \cot x| \). Using the Pythagorean identity \( \csc^2 x - \cot^2 x = 1 \), we can express \( \csc x - \cot x \) and \( \csc x + \cot x \) as conjugates such that their product is 1: \((\csc x - \cot x)(\csc x + \cot x) = (\csc^2 x - \cot^2 x) = 1\).

Step 3: Use Logarithmic Properties

Utilize the logarithmic property that \( \ln(a) + \ln(b) = \ln(ab) \). Therefore, if \( \csc x - \cot x \) and \( \csc x + \cot x \) are such that \((\csc x - \cot x)(\csc x + \cot x) = 1\), then \( \ln |\csc x - \cot x| + \ln |\csc x + \cot x| = \ln 1 = 0 \).

Step 4: Rearrange the Expression

From Step 3, we have \( \ln |\csc x - \cot x| + \ln |\csc x + \cot x| = 0 \). Rearrange this to find \( \ln |\csc x - \cot x| = -\ln |\csc x + \cot x| \), which matches the given identity.

Step 5: Conclusion: Verify the Identity

We showed that the expression \( \ln |\csc x - \cot x| = -\ln |\csc x + \cot x| \) is true by utilizing properties of logarithms and trigonometric identities.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that relate the angles and lengths of triangles, primarily for the six trigonometric functions: sine (\( \sin \)), cosine (\( \cos \)), tangent (\( \tan \)), cosecant (\( \csc \)), secant (\( \sec \)), and cotangent (\( \cot \)). These identities are essential for simplifying complex trigonometric expressions and solving equations.

• **Basic Identities**: These include \( \sin^2 x + \cos^2 x = 1 \), \( \tan x = \frac{\sin x}{\cos x }\), and \( \csc x = \frac{1}{\sin x}\).
• **Reciprocal Identities**: Such as \( \csc x = \frac{1}{\sin x} \) and \( \cot x = \frac{1}{\tan x} \).
• **Pythagorean Identities**: These are extensions of the Pythagorean theorem, like \( \csc^2 x - \cot^2 x = 1 \).

Understanding these identities allows for the simplification and transformation of expressions, which is crucial in verifying the given logarithmic identity.

Pythagorean Identity

The Pythagorean identity refers to a set of fundamental equations that are derived from the Pythagorean theorem. Most commonly known is:\[ \sin^2 x + \cos^2 x = 1 \]For cosecant and cotangent, we rely on the identity:\[ \csc^2 x - \cot^2 x = 1 \]This identity can be viewed as a rearrangement of the fundamental Pythagorean theorem applied to the unit circle.
- **Application**: This identity is useful for transforming trigonometric expressions and plays a key role in our example where we expand \((\csc x - \cot x)(\csc x + \cot x)\) into 1, verifying the original identity.It demonstrates how expressions involving trigonometric functions can multiply to give simple values, facilitating easier manipulation and simplification of equations.

Cosecant and Cotangent

The cosecant (\( \csc x \)) and cotangent (\( \cot x \)) functions are reciprocal trigonometric functions.

• **Cosecant Function**: Defined as \( \csc x = \frac{1}{\sin x} \). It is the reciprocal of the sine function.
• **Cotangent Function**: Defined as \( \cot x = \frac{1}{\tan x} \) or equivalently \( \cot x = \frac{\cos x}{\sin x} \). It is the reciprocal of the tangent function.

In the identity \( \ln |\csc x - \cot x| = -\ln |\csc x + \cot x| \), these functions are combined in difference and sum forms under absolute values. Understanding their properties allows us to manipulate these terms effectively.
Their interplay through identities like the Pythagorean identity aids in establishing that their product \((\csc x - \cot x)(\csc x + \cot x) = 1\), crucial for the logarithmic property used in the solution.

Properties of Logarithms

Logarithms have several properties that are crucial for solving equations involving them.

• **Product Property**: \( \ln(a) + \ln(b) = \ln(ab) \). This property states that the log of a product is the sum of the logs.
• **Identity for Zero**: \( \ln 1 = 0 \). Since any number to the power of 0 is 1, the logarithm of 1 is always 0.
• **Negative Property**: \( \ln(a) = -\ln(\frac{1}{a}) \). This property shows that the logarithm of a reciprocal is the negative of the logarithm.

In our verification task, the product property is essential. It allows us to combine the logarithms into a single expression, ultimately demonstrating that the sum of the two expressions is zero, thus proving the identity.