Problem 74 - Precalculus with Calculus Previews

Question

In Problems \(73-76\), verify the given identity. $$ \ln |\cot x|=-\ln |\tan x| $$

Step-by-Step Solution

Answer

The identity \( \ln |\cot x| = -\ln |\tan x| \) is verified using logarithm properties and trigonometric identities.

Step 1: Understand the logarithm properties

Recall that the logarithm function has several properties. One key property is that \( \ln |a| = - \ln |\frac{1}{a}| \). This property will be useful in transforming one side of the identity into the other.

Step 2: Express \( \ln |\tan x| \) using trigonometric identities

The tangent function can be expressed as the reciprocal of the cotangent function. Specifically, \( \tan x = \frac{1}{\cot x} \). Therefore, taking the logarithm of the absolute value, \( \ln |\tan x| = \ln \left| \frac{1}{\cot x} \right| \).

Step 3: Apply the logarithmic property

Using the property \( \ln |a| = - \ln |\frac{1}{a}| \), we can rewrite \( \ln \left| \frac{1}{\cot x} \right| = - \ln |\cot x| \).

Step 4: Verify the identity

Substitute back to see if both sides of the original identity are equal. Starting with the left-hand side \( \ln |\cot x| \) and using our transformation, we obtain the right-hand side \( -\ln |\tan x| \). Thus, the identity \( \ln |\cot x| = -\ln |\tan x| \) is verified.

Key Concepts

Trigonometric Identities

Trigonometric identities are mathematical equations that express relationships between trigonometric functions such as sine, cosine, and tangent. These identities are especially useful in simplifying complex trigonometric expressions. They help us transform an equation into a more traditionally solvable form.
A well-known trigonometric identity involves the relationship between tangent and cotangent functions. The tangent function, denoted as \( \tan x \), is the ratio of sine \( \sin x \) to cosine \( \cos x \), meaning \( \tan x = \frac{\sin x}{\cos x} \). The cotangent, \( \cot x \), is the reciprocal of the tangent, given by \( \cot x = \frac{\cos x}{\sin x} \).
When exploring logarithmic expressions, these identities can help us express and simplify the expressions, like in our original problem. This mathematical convenience is important when using trigonometric identities to solve such equations efficiently.

Reciprocal Functions

Reciprocal functions play a crucial role in algebra and trigonometry. A reciprocal of a function is another function which stands for the division of one by the initial function. For example, if \( f(x) = a \), then its reciprocal is \( \frac{1}{f(x)} = \frac{1}{a} \). These functions simplify the operations of division into expressions that only involve multiplication.
In trigonometry, the cotangent and tangent functions are reciprocals. If \( \tan x = \frac{\sin x}{\cos x} \), then \( \cot x = \frac{\cos x}{\sin x} \), making \( \cot x = \frac{1}{\tan x} \).

• This reciprocal relation is essential because it helps to transform arguments when working with identities, allowing us to make conversions between \( \tan x \) and \( \cot x \).
• Using reciprocals simplifies solving identities, as we can switch between related functions effortlessly.

In essence, by understanding reciprocal relationships, we manage to connect different trigonometric functions and facilitate the process of solving problems like the one presented.

Properties of Logarithms

Logarithmic properties are foundational to simplifying and understanding complex logarithmic expressions. One of the most crucial properties is the change of base, which allows us to express logarithms in terms of other bases or the numbers themselves.
The key property we use in verifying the identity \( \ln |\cot x| = - \ln |\tan x| \) is that \( \ln |a| = - \ln |\frac{1}{a}| \). This property indicates that the logarithm of a reciprocal translates into a negative version of the original logarithm.

• For example, if \( \ln a \) is the natural logarithm of \( a \), then \( \ln \frac{1}{a} = - \ln a \).
• This relationship allows simplification of expressions involving division and ensures consistency across logarithmic manipulations.

By applying this property, we can verify identities by transforming complex or cumbersome expressions into simpler equivalents. Understanding these properties ensures effective problem-solving and aids in comprehension of deeper mathematical concepts.