Problem 24 - Algebra 2 and Trigonometry

Question

In \(3-26,\) prove that each equation is an identity. $$ \frac{\sec \theta}{\cos \theta}-\tan ^{2} \theta=1 $$

Step-by-Step Solution

Answer

The given equation is an identity, verified using trigonometric identities and simplifications.

Step 1: Rewrite trigonometric functions

Recall the definitions of secant and tangent in terms of sine and cosine. We know that \( \sec \theta = \frac{1}{\cos \theta} \) and \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) so \( \tan^2 \theta = \left(\frac{\sin \theta}{\cos \theta}\right)^2 = \frac{\sin^2 \theta}{\cos^2 \theta} \).

Step 2: Use trigonometric identities

Use the Pythagorean identity: \( \sin^2 \theta + \cos^2 \theta = 1 \). Rearranging gives \( \sin^2 \theta = 1 - \cos^2 \theta \).

Step 3: Substitute into the original equation

Substitute \( \sec \theta = \frac{1}{\cos \theta} \) and \( \tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta} \) into the original equation:\[ \frac{\sec \theta}{\cos \theta} - \tan^2 \theta = 1 \quad \Rightarrow \quad \frac{\frac{1}{\cos \theta}}{\cos \theta} - \frac{\sin^2 \theta}{\cos^2 \theta} = 1 \]

Step 4: Simplify fractions

Simplify \( \frac{\frac{1}{\cos \theta}}{\cos \theta} \) to get \( \frac{1}{\cos^2 \theta} \). Substituting this back gives us:\[ \frac{1}{\cos^2 \theta} - \frac{\sin^2 \theta}{\cos^2 \theta} = 1 \]

Step 5: Combine fractions

Since both terms on the left side have the same denominator, combine them: \[ \frac{1 - \sin^2 \theta}{\cos^2 \theta} = 1 \]

Step 6: Use Pythagorean identity again

Substitute \( 1 - \sin^2 \theta \) with \( \cos^2 \theta \) using the Pythagorean identity. This transforms the equation into:\[ \frac{\cos^2 \theta}{\cos^2 \theta} = 1 \]

Step 7: Conclude the proof

Since \( \frac{\cos^2 \theta}{\cos^2 \theta} = 1 \) is clearly true, we have proven that the original equation is an identity.

Key Concepts

Secant

The secant function, denoted as \( \sec \theta \), is one of the fundamental trigonometric functions, often introduced alongside sine, cosine, and tangent. Its relationship with the cosine function is crucial, as \( \sec \theta \) is defined as the reciprocal of the cosine of \( \theta \). This means:

• \( \sec \theta = \frac{1}{\cos \theta} \)

This reciprocal characteristic allows secant to provide useful information in situations where cosine is zero because the function alone, \( \cos \theta = 0 \), would make another form of the equation undefined. By transforming expressions using secant, many complex trigonometric problems can be simplified and better understood. When working through proofs or trying to simplify trigonometric equations, recognizing this definition can be incredibly helpful. As we express other functions, like tangent, in terms of sine and cosine, keeping an eye on how secant interacts with these can further simplify our work.

Tangent

Tangent, denoted by \( \tan \theta \), is a crucial trigonometric function that compares the relationship between sine and cosine. Specifically, it is defined as the ratio of the sine to the cosine of an angle \( \theta \). Mathematically, this relationship can be written as follows:

• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

By squaring tangent, we get a different perspective, \( \tan^2 \theta \), essential for working through various trigonometric identities. This square can be expanded using the underlying definitions:

• \( \tan^2 \theta = \left(\frac{\sin \theta}{\cos \theta}\right)^2 = \frac{\sin^2 \theta}{\cos^2 \theta} \)

Using tangent in conjunction with secant not only highlights how these functions interrelate but also illustrates how identities can be creatively rearranged and simplified for solving equations or proving complex relationships.

Pythagorean Identity

The Pythagorean identity is a fundamental aspect in studying trigonometry, pivotal for proving many other identities and equations. The foundational form of this identity is:

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

This identity provides a powerful relationship between sine and cosine, encapsulating their squared functions' essence. It's especially useful for transforming complex expressions involving trigonometric functions. For instance, rearranging the basic identity gives us:

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

Using such rearrangements, like in the problem at hand, often turns challenging expressions into manageable forms. Whenever faced with intricate trigonometric expressions, leveraging the Pythagorean identity can be the key to unlock simplifications and understand a problem's deeper structure.

Trigonometric Simplification

Simplifying trigonometric equations can turn seemingly complex problems into manageable tasks. The goal is often to express all terms using common denominators or easily convertible identities. Consider the equation:

• \( \frac{\sec \theta}{\cos \theta} - \tan^2 \theta = 1 \)

The original method involves substituting definitions such as \( \sec \theta = \frac{1}{\cos \theta} \) and \( \tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta} \). Through simplification, the left-hand side becomes:

• \( \frac{1}{\cos^2 \theta} - \frac{\sin^2 \theta}{\cos^2 \theta} \)

By utilizing common denominators, these terms can be combined. Leveraging the Pythagorean identity \( 1 - \sin^2 \theta = \cos^2 \theta \) allows this expression to be recognized as a structural identity, confirming its truth. Mastery in trigonometric simplification, recognizing when and how to apply identities, turns otherwise intimidating problems into straightforward solutions. This process reveals not just the solution to a problem but the harmonious balance inherent in trigonometric relationships.