Problem 56 - Precalculus with Calculus Previews

Question

Verify the given identity. $$ \sec 2 x=\frac{1}{2 \cos ^{2} x-1} $$

Step-by-Step Solution

Answer

The identity is verified by using the double angle identity for cosine.

Step 1: Understanding the Identity

We need to verify the trigonometric identity \( \sec 2x = \frac{1}{2\cos^2 x - 1} \). This means we have to show that the left side of the equation is equal to the right side of the equation for any angle \( x \).

Step 2: Express \( \sec 2x \) Using Trigonometric Function

Start by expressing \( \sec 2x \) as \( \frac{1}{\cos 2x} \) since \( \sec\theta = \frac{1}{\cos\theta} \). Therefore, \( \sec 2x = \frac{1}{\cos 2x} \).

Step 3: Use Double Angle Identity for Cosine

The double angle identity for cosine states \( \cos 2x = 2\cos^2 x - 1 \). Substitute \( \cos 2x \) with this identity in \( \sec 2x = \frac{1}{\cos 2x} \). This gives \( \sec 2x = \frac{1}{2\cos^2 x - 1} \).

Step 4: Verification

We have rewritten \( \sec 2x \) as \( \frac{1}{2\cos^2 x - 1} \) using the double angle identity. This matches the right side of our given identity, thus the identity is verified.

Key Concepts

Double Angle Formulas

Double angle formulas are a set of trigonometric identities that provide a way to express trigonometric functions of double angles in terms of single angles. These identities are particularly useful because they allow for simplification and transformation of trigonometric expressions. For instance, the double angle formula for cosine is given by:\[ \cos(2x) = 2\cos^2(x) - 1 \]This formula can be derived from the sum identities and is especially handy when dealing with equations involving double angles like in our exercise.

• Cosine: The formula \( \cos(2x) = 2\cos^2 x - 1 \) helps us convert expressions involving \( \cos 2x \) into ones involving \( \cos x \) only, simplifying complex identities.
• Sine and Tangent: There are similar formulas for sine, \( \sin(2x) = 2\sin x \cos x \), and tangent, \( \tan(2x) = \frac{2\tan x}{1 - \tan^2 x} \), each derived from their basic trigonometric properties.

These formulas help in breaking down identities into simpler components, making it easier to solve or verify them.

Verification of Identities

Verification of trigonometric identities involves proving that one side of an equation equals the other by using known trigonometric identities and formulas. It's a crucial skill in trigonometry that enhances understanding and problem-solving abilities.
To verify an identity such as \( \sec 2x = \frac{1}{2\cos^2 x - 1} \), follow these steps:

• Step 1: Rewrite complex expressions using fundamental identities like \( \sec\theta = \frac{1}{\cos\theta} \). For our case, rewriting the secant function gives \( \sec 2x = \frac{1}{\cos 2x} \).
• Step 2: Apply the appropriate trigonometric identity or formula, such as the double angle formula for cosine, \( \cos 2x = 2\cos^2 x - 1 \), to transform and simplify one side of the equation.
• Step 3: Simplify each side systematically until they are identical. This confirms the identity, as demonstrated where both sides equal \( \frac{1}{2\cos^2 x - 1} \).

Verification exercises improve your ability to manipulate and understand different trigonometric expressions.

Trigonometric Functions

Trigonometric functions form the foundation of many mathematical concepts and are essential for understanding angles and their relationships. They are based on the relationships of angles and sides within a right-angled triangle and are broadly categorized as sine, cosine, tangent, secant, cosecant, and cotangent.

• Sine and Cosine: These are the primary trigonometric functions from which other functions are derived. Sine, \( \sin \theta \), represents the ratio of the opposite side to the hypotenuse, while cosine, \( \cos \theta \), is the ratio of the adjacent side to the hypotenuse in a right triangle.
• Tangent: Defined as the ratio of sine to cosine, \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).
• Secant: The reciprocal of cosine, given by \( \sec \theta = \frac{1}{\cos \theta} \).

Each function has its own specific properties and applications, interrelating through various identities. In our exercise, the focus on secant and cosine demonstrates how these identities are manipulated for verification. Understanding these functions is key to mastering trigonometry, enabling easier manipulation and transformation of expressions and equations.