Problem 49 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-50: Verify the identity. $$ \ln |\sec \theta+\tan \theta|=-\ln |\sec \theta-\tan \theta| $$

Step-by-Step Solution

Answer

The identity is verified.

Step 1: Apply properties of logarithms

Let's start with the left-hand side of the given identity, \( \ln |\sec \theta + \tan \theta| \). According to the problem, this should equal \( -\ln |\sec \theta - \tan \theta| \). Recall the logarithmic property \( -\ln x = \ln \left(\frac{1}{x}\right) \). We can apply this to the right-hand side, so:\[ \ln |\sec \theta + \tan \theta| = \ln \left( \frac{1}{|\sec \theta - \tan \theta|} \right). \] We need to prove that:\[ |\sec \theta + \tan \theta| = \frac{1}{|\sec \theta - \tan \theta|}. \]

Step 2: Simplify the trigonometric expression

To verify the identity \( |\sec \theta + \tan \theta| = \frac{1}{|\sec \theta - \tan \theta|} \), we need to work with trigonometric identities. Recall the identity for secant and tangent:\[ \sec \theta = \frac{1}{\cos \theta}, \quad \tan \theta = \frac{\sin \theta}{\cos \theta}. \]Substituting these, we have:\[ \sec \theta + \tan \theta = \frac{1}{\cos \theta} + \frac{\sin \theta}{\cos \theta} = \frac{1 + \sin \theta}{\cos \theta}, \]\[ \sec \theta - \tan \theta = \frac{1}{\cos \theta} - \frac{\sin \theta}{\cos \theta} = \frac{1 - \sin \theta}{\cos \theta}. \]

Step 3: Explore the relationship between the expressions

Now let's explore the relationship between the expressions derived previously:\[ \frac{1}{|\sec \theta - \tan \theta|} = \left| \frac{1}{\frac{1 - \sin \theta}{\cos \theta}} \right| = \left| \frac{\cos \theta}{1 - \sin \theta} \right|. \]On the other hand, the expression for \( |\sec \theta + \tan \theta| \) was:\[ |\sec \theta + \tan \theta| = \left| \frac{1 + \sin \theta}{\cos \theta} \right|. \]

Step 4: Equate and simplify both sides

To verify the identity, equate the expressions found:\[ \left| \frac{1 + \sin \theta}{\cos \theta} \right| = \left| \frac{\cos \theta}{1 - \sin \theta} \right|. \]By simplifying these, we multiply and divide each side appropriately:\[ \left| \frac{(1 + \sin \theta)(1 - \sin \theta)}{\cos \theta \cdot \cos \theta} \right| = 1. \]Using the Pythagorean identity, \( (1 + \sin \theta)(1 - \sin \theta) = 1 - \sin^2 \theta = \cos^2 \theta \), the equation becomes:\[ \left| \frac{\cos^2 \theta}{\cos^2 \theta} \right| = 1, \]which is true, verifying the identity.

Key Concepts

Logarithmic Properties

Logarithmic properties are essential tools in mathematics, particularly when simplifying and solving equations involving logarithms. The primary property used in this exercise is the one involving negative logarithms:

• Negative Logarithm Property: \( -\ln x = \ln \left(\frac{1}{x}\right) \)

This property transforms the negative sign into the reciprocal of the logarithmic argument. By using this transformation, we can manipulate expressions to make them more manageable or to match them to an equivalent form.
In our problem, this property allows us to verify that the expressions on both sides of the identity are equivalent, by changing the sign to a reciprocal—a simple yet powerful transformation in logarithmic equations.

Trigonometric Identities

Trigonometric identities are formulas involving trigonometric functions that are true for every value of the variable where both sides of the identity are defined. Two key trigonometric identities used in this exercise include:

• Secant Identity: \( \sec \theta = \frac{1}{\cos \theta} \)
• Tangent Identity: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

These identities allow us to express secant and tangent in terms of sine and cosine. By rewriting \( \sec \theta + \tan \theta \) and \( \sec \theta - \tan \theta \) in terms of sine and cosine, complicated expressions can become much simpler. This transformation is crucial for verifying and proving the given identity as it directly leads to equivalent expressions that can be compared more conveniently.

Secant and Tangent

Focusing on secant \( (\sec \theta) \) and tangent \( (\tan \theta) \) is pivotal when dealing with trigonometric identities.
Both functions are defined through sine and cosine, and their definitions help in simplifying expressions:

• Secant: This is the reciprocal of cosine, \( \sec \theta = \frac{1}{\cos \theta} \). It represents the ratio of the hypotenuse to the adjacent side in a right triangle.
• Tangent: This is the ratio of sine to cosine, \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). It represents the ratio of the opposite side to the adjacent side in a right triangle.

Understanding these relationships helps to further breakdown and rearrange expressions involving secant and tangent, which is essential in our problem where the original expression \( \ln |\sec \theta + \tan \theta| \) needs to equal \( -\ln |\sec \theta - \tan \theta| \). By using their definitions, it's easier to show their equivalence.

Pythagorean Identity

The Pythagorean identity is one of the most fundamental identities in trigonometry and states that for any angle \( \theta \), the following is true:

• \( \sin^2 \theta + \cos^2 \theta = 1 \)

This identity is key to many derivations and proofs in trigonometry as it relates directly to the unit circle. In the verification of our identity, a rearranged version of this identity plays a crucial role.
When we have \( (1 + \sin \theta)(1 - \sin \theta) \), it simplifies to \( 1 - \sin^2 \theta \), which is equal to \( \cos^2 \theta \) because of the Pythagorean identity. This step is necessary because it simplifies the expression \( \left| \frac{(1 + \sin \theta)(1 - \sin \theta)}{\cos^2 \theta} \right| \) to 1.
Using this identity confirms that the simplifications and transformations carried out align with the fundamental truths of trigonometric functions, thereby verifying the given trigonometric identity.