Problem 50 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-50: Verify the identity. $$ \ln |\csc x-\cot x|=-\ln |\csc x+\cot x| $$

Step-by-Step Solution

Answer

The identity is verified by showing both sides are reciprocals.

Step 1: Understand the Identity

We are given the identity to verify: \( \ln |\csc x - \cot x| = -\ln |\csc x + \cot x| \). This means we need to show that these two expressions are equivalent, or the negative of each other, using mathematical properties and transformations.

Step 2: Apply Logarithm Property

Notice that if we have \( \ln a = -\ln b \), then we can deduce \( a = b^{-1} \). Therefore, we aim to show that \( |\csc x - \cot x| = (|\csc x + \cot x|)^{-1} \).

Step 3: Use Trigonometric Identities

Recall the definitions of cosecant and cotangent: \( \csc x = \frac{1}{\sin x} \) and \( \cot x = \frac{\cos x}{\sin x} \). Substitute these into the expressions: \(|\csc x - \cot x| = \left| \frac{1 - \cos x}{\sin x} \right|\) and \(|\csc x + \cot x| = \left| \frac{1 + \cos x}{\sin x} \right|\).

Step 4: Simplify the Expressions

Now simplify the expressions: \( |\csc x - \cot x| = \left| \frac{1 - \cos x}{\sin x} \right| = \left| \frac{(1 - \cos x)(1 + \cos x)}{(1 + \cos x)\sin x} \right| = \left| \frac{1 - \cos^2 x}{(1 + \cos x)\sin x} \right| = \left| \frac{\sin^2 x}{(1 + \cos x)\sin x} \right| \). Thus, this becomes \( |\sin x| |1 - \cos x| \).

Step 5: Show the Reciprocal

The expression for \( |\csc x + \cot x| \) was simplified to \( |\sin x| |1 + \cos x| \). By comparing, the two expressions obtained are reciprocals. Specifically, \(|\csc x - \cot x| = |\sin x| |1 - \cos x|\) and its reciprocal is \((|\csc x + \cot x|)^{-1} = \frac{1}{|\sin x| |1 + \cos x|}\). They match the reciprocal relationship, proving the original identity.

Key Concepts

Logarithmic Identities

Logarithmic identities are mathematical statements that simplify the manipulation of logarithms. One essential property is that if you have

• \( \ln a = -\ln b \),

you can infer that \( a = b^{-1} \). This is because logarithms can transform multiplicative expressions into additive ones, revealing hidden relationships.

In our exercise, the logarithm identity helps to show how two expressions can be negatives of each other in logarithmic terms. Specifically,

• \( \ln |\csc x - \cot x| = -\ln |\csc x + \cot x| \)

can be rewritten as

• \( |\csc x - \cot x| = (|\csc x + \cot x|)^{-1} \).

This means that the expressions inside the logarithm are reciprocal values.

Cosecant

The cosecant function, denoted as \( \csc x \), is one of the trigonometric functions. It is defined as the reciprocal of the sine function. Thus,

• \( \csc x = \frac{1}{\sin x} \).

This relationship means that wherever sine is zero, cosecant is undefined since you cannot divide by zero.

In our exercise, this definition is used to express the terms within the logarithms. When transformed using the definition,

• \( |\csc x - \cot x| \)
• and \( |\csc x + \cot x| \)

become differences and sums of fractional expressions, which can be simplified further to prove their reciprocal nature.

Cotangent

The cotangent function, abbreviated as \( \cot x \), is another trigonometric function. It is the reciprocal of the tangent function, which is defined as the ratio of the cosine over the sine of an angle.

• Thus, \( \cot x = \frac{\cos x}{\sin x} \).

Since \( \cot x \) is expressed in terms of \( \sin x \) and \( \cos x \), it assists in transforming trigonometric identities by incorporating cosine values. In our identity verification exercise,

• \( \csc x - \cot x \)
• and \( \csc x + \cot x \)

are simplified using the above definition. Together with \( \csc x \), cotangent helps reveal a relationship that simplifies to show a reciprocal condition.

Reciprocal

Reciprocal relationships are fundamental in math, implying that two quantities multiply to one. A reciprocal of a number \( a \) is \( a^{-1} \), which, when multiplied by \( a \), equals 1.

• For example, the reciprocal of \( 2 \) is \( \frac{1}{2} \).

In trigonometry, intriguing reciprocal identities often arise. For instance, the reciprocal of cosecant is sine. In our exercise,

• the connection \( |\csc x - \cot x| \)
• with \( |\csc x + \cot x| \)

is demonstrated through reciprocal trigonometric simplification. We simplify both expressions and show they multiply to yield 1, thus confirming their reciprocal relationship and verifying the logarithmic identity.