Problem 49 - Precalculus: Functions and Graphs

Question

Verify the Identity. $$\ln |\sec \theta+\tan \theta|=-\ln |\sec \theta-\tan \theta|$$

Step-by-Step Solution

Answer

The identity is verified as both sides are reciprocals.

Step 1: Understand the Identity

The given identity to verify is \( \ln |\sec \theta + \tan \theta| = -\ln |\sec \theta - \tan \theta| \). This means we need to show that the logarithm of the left side is the negative of the logarithm of the right side.

Step 2: Use Logarithmic Properties

We know from the properties of logarithms that \( \ln(a) = -\ln(b) \) implies \( a = \frac{1}{b} \). Thus, we need to verify if \( \sec \theta + \tan \theta = \frac{1}{\sec \theta - \tan \theta} \).

Step 3: Express in Terms of Sine and Cosine

Recall the definitions: \( \sec \theta = \frac{1}{\cos \theta} \) and \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Substitute these into the equation to rewrite both expressions.

Step 4: Simplify the Left Side

The expression \( \sec \theta + \tan \theta \) becomes \( \frac{1}{\cos \theta} + \frac{\sin \theta}{\cos \theta} = \frac{1 + \sin \theta}{\cos \theta} \).

Step 5: Simplify the Right Side

The expression \( \sec \theta - \tan \theta \) becomes \( \frac{1}{\cos \theta} - \frac{\sin \theta}{\cos \theta} = \frac{1 - \sin \theta}{\cos \theta} \).

Step 6: Invert the Right Side

Find the reciprocal of \( \frac{1 - \sin \theta}{\cos \theta} \), which is \( \frac{\cos \theta}{1 - \sin \theta} \).

Step 7: Verify Equality

Compare the expression from Step 4 (left side) \( \frac{1 + \sin \theta}{\cos \theta} \) with the reciprocal of Step 5 (right side) \( \frac{\cos \theta}{1 - \sin \theta} \). Simplify to see if they are equal.

Step 8: Verify Completion

Cross multiply \( (1 + \sin \theta)(1 - \sin \theta) = \cos^2 \theta \). The left side simplifies to \( 1 - \sin^2 \theta \), which equals \( \cos^2 \theta \). Hence, they are equal, verifying the identity.

Key Concepts

Logarithmic Properties

Logarithmic properties are incredibly useful when manipulating expressions. One important property you often encounter is that if \( \ln(a) = -\ln(b) \), it implies that \( a \) and \( b \) are reciprocals. Specifically, this suggests that \( a = \frac{1}{b} \).

This property arises from the power rule of logarithms, which states that \( \ln(a^{-1}) = -\ln(a) \). Therefore, in problems such as verifying the identity \( \ln |\sec \theta + \tan \theta| = -\ln |\sec \theta - \tan \theta| \), understanding that the expressions are inverses of each other helps in simplifying the solution process.

• **Reciprocal Relationship:** Recognizing that \( a^{-1} = \frac{1}{a} \) helps to equate the two expressions.
• **Application:** Use these logarithmic properties to simplify the problem, turning a complex expression into a more manageable form.
• **Verification:** Once simplifications are complete, confirm that the structure adheres to those logarithmic rules.

This technique aids in solving a myriad of mathematical problems by reducing the complexity of logarithmic equations into something more digestible.

Sine and Cosine Definitions

Understanding the definitions of sine \( \sin \theta \) and cosine \( \cos \theta \) is crucial for solving trigonometric identities.

The sine of an angle \( \theta \) in a right triangle is the ratio of the opposite side to the hypotenuse, and cosine is the ratio of the adjacent side to the hypotenuse. This fundamental context explains why \( \sec \theta = \frac{1}{\cos \theta} \) and \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).

• **Basic Definitions:** Sine and cosine are fundamental to trigonometry, stemming from geometric principles.
• **Relation to Other Functions:** Recognizing \( \tan \theta \) and \( \sec \theta \) in terms of sine and cosine helps simplify expressions.
• **Solving Identities:** By substituting sine and cosine definitions, you can transform expressions into more straightforward versions to simplify and solve them.

When faced with identities, utilizing these basic definitions allows you to break down more complex relationships into known quantities.

Reciprocal Identities

Reciprocal identities are a fundamental aspect of trigonometry and assist in transforming and simplifying trigonometric expressions. Knowing the reciprocals of the trigonometric functions is key.

The reciprocal of secant, \( \sec \theta \), is cosine, as \( \sec \theta = \frac{1}{\cos \theta} \), making it \( \cos \theta = \frac{1}{\sec \theta} \). Similarly, for tangent, \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), its reciprocal function is cotangent, \( \cot \theta = \frac{1}{\tan \theta} \).

• **Understanding Reciprocals:** Each trigonometric function has a counterpart, influencing how they are manipulated.
• **Application in Identities:** Applied correctly, reciprocal identities simplify complex trigonometric problems.
• **Integration with Definitions:** Use reciprocal identities in conjunction with sine and cosine definitions to approach various trigonometric identities.

By mastering reciprocal identities, you equip yourself with tools necessary for simplifying and resolving intricate trigonometric equations.