Problem 47 - Precalculus: Functions and Graphs

Question

Verify the Identity. $$\ln \cot x=-\ln \tan x$$

Step-by-Step Solution

Answer

The identity \( \ln \cot x = -\ln \tan x \) is verified as both sides simplify to \( \ln \cos x - \ln \sin x \).

Step 1: Express Cotangent and Tangent in Terms of Sine and Cosine

We know that \( \cot x = \frac{\cos x}{\sin x} \) and \( \tan x = \frac{\sin x}{\cos x} \). Replacing \( \cot x \) in the equation gives \( \ln \left( \frac{\cos x}{\sin x} \right) \).

Step 2: Use the Logarithm Property on Cotangent

Using the property \( \ln \left( \frac{a}{b} \right) = \ln a - \ln b \), we can rewrite \( \ln \left( \frac{\cos x}{\sin x} \right) \) as \( \ln \cos x - \ln \sin x \).

Step 3: Rewrite the Right Side Using Tangent Expression

Remembering that \( \tan x = \frac{\sin x}{\cos x} \), substitute into \( -\ln \tan x \) to get \( -\ln \left( \frac{\sin x}{\cos x} \right) \).

Step 4: Apply Logarithm Property to the Negation

Using the logarithm property, \( -\ln \left( \frac{\sin x}{\cos x} \right) = - (\ln \sin x - \ln \cos x) \), which simplifies to \( -\ln \sin x + \ln \cos x \).

Step 5: Compare Both Sides of the Identity

Both sides simplify to \( \ln \cos x - \ln \sin x \), verifying that \( \ln \cot x = -\ln \tan x \) is true.

Key Concepts

Trigonometric Identities

Trigonometric identities are mathematical statements that equate one expression to another in terms of trigonometric functions. They are essential tools in simplifying complex expressions and solving trigonometric equations. To understand these identities, consider some fundamental ones like the Pythagorean identity, which states \(\sin^2 x + \cos^2 x = 1\). This helps in transforming and verifying expressions.

In the exercise, the identity being verified involves \(\cot x\) and \(\tan x\), which are the reciprocals of tangent and cotangent functions respectively. The goal here is to express the original identity \(\ln \cot x = -\ln \tan x\) in alternative formats that reveal their equivalence.

• The cotangent function, \(\cot x\), is expressed as \(\frac{\cos x}{\sin x}\).
• The tangent function, \(\tan x\), is expressed as \(\frac{\sin x}{\cos x}\).

These identities show the interplay between sine and cosine, making them the building blocks to prove the given logarithmic identity.

Properties of Logarithms

Logarithms have specific properties that make them incredibly useful for simplifying expressions. These properties allow operations such as multiplication, division, and powers to be expressed in a simplified logarithmic form. Let's explore a couple of key properties:

• Product Property: \(\ln ab = \ln a + \ln b\)
• Quotient Property: \(\ln \left(\frac{a}{b}\right) = \ln a - \ln b\)
• Power Property: \(\ln a^b = b \ln a\)

In the original step-by-step solution, the quotient property is crucial. It's used to break down the expression \(\ln \left(\frac{\cos x}{\sin x}\right)\) into \(\ln \cos x - \ln \sin x\). Understanding these properties forms a foundation for manipulating logarithmic expressions, as shown in verifying the exercise identity.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, tangent, cotangent, secant, and cosecant, describe the relationships between the angles and sides of a right triangle. They are periodic, which means they repeat their values in regular intervals, making them cornerstone functions in trigonometry.

Each trigonometric function can be defined using a unit circle. For instance, the sine and cosine functions are linked to the vertical and horizontal positions of a point rotating around the circle. These functions are fundamental in expressing other trigonometric functions:

• \(\sin x = \text{opposite} / \text{hypotenuse}\) in a right triangle.
• \(\cos x = \text{adjacent} / \text{hypotenuse}\) in a right triangle.
• \(\tan x = \sin x / \cos x\)
• \(\cot x = 1 / \tan x = \cos x / \sin x\)

In the context of verifying the identity \(\ln \cot x = -\ln \tan x\), understanding these functions in terms of sine and cosine is vital. They allow us to transform \(\cot x\) and \(\tan x\) into ratios of sine and cosine, providing clarity in why the logarithmic expressions on both sides of the identity are equivalent.