Problem 23 - Algebra and Trigonometry with Analytic Geometry

Question

Verify the identity. $$ \sec 2 \theta=\frac{\sec ^{2} \theta}{2-\sec ^{2} \theta} $$

Step-by-Step Solution

Answer

The identity holds true: \( \sec 2\theta = \frac{\sec^2(\theta)}{2-\sec^2(\theta)} \).

Step 1: Recall Identity of Secant Function

First, recall that the function \( \sec(\theta) = \frac{1}{\cos(\theta)} \). This identity will be useful in transforming parts of the given expression.

Step 2: Use Double Angle Identity for Secant

The double angle identity for secant is \( \sec(2\theta) = \frac{1}{\cos(2\theta)} \). According to the double angle identity for cosine, \( \cos(2\theta) = 2\cos^2(\theta) - 1 \). Therefore, \( \sec(2\theta) = \frac{1}{2\cos^2(\theta) - 1} \).

Step 3: Simplify the Right Side

The right-hand side is \( \frac{\sec^2(\theta)}{2 - \sec^2(\theta)} \). Replace \( \sec^2(\theta) \) with \( \frac{1}{\cos^2(\theta)} \), so this becomes \( \frac{\frac{1}{\cos^2(\theta)}}{2 - \frac{1}{\cos^2(\theta)}} \).

Step 4: Simplify the Denominator

Combine terms in the denominator: \( 2 - \frac{1}{\cos^2(\theta)} = \frac{2\cos^2(\theta) - 1}{\cos^2(\theta)} \).

Step 5: Divide Fractions

The expression becomes \( \frac{\frac{1}{\cos^2(\theta)}}{\frac{2\cos^2(\theta) - 1}{\cos^2(\theta)}} \). Dividing fractions is the same as multiplying by the reciprocal, so this becomes \( \frac{1}{\cos^2(\theta)} \times \frac{\cos^2(\theta)}{2\cos^2(\theta) - 1} \), which simplifies to \( \frac{1}{2\cos^2(\theta) - 1} \).

Step 6: Compare Both Sides

We previously found \( \sec(2\theta) = \frac{1}{2\cos^2(\theta) - 1} \). The simplified form of the right-hand side is also \( \frac{1}{2\cos^2(\theta) - 1} \). Therefore, both sides of the original identity are equal.

Key Concepts

Secant Function

The secant function, denoted as \( \sec \theta \), is an essential part of trigonometry. It is the reciprocal of the cosine function. In formula terms, it is expressed as:
\[ \sec \theta = \frac{1}{\cos \theta} \]This equation is particularly helpful because it allows us to transform expressions involving secant into ones involving cosine, making calculations more manageable. Given these characteristics, the secant function allows for various manipulations and simplifications in trigonometric identities and equations.
Whenever you encounter a secant, remember its close relationship with cosine. This can be extremely useful in breaking down more complex trigonometric identities and understanding their behavior.

Double Angle Identities

Double angle identities are remarkably useful in trigonometry, especially when you want to express a trigonometric function of\( 2\theta \) in terms of \( \theta \). For the secant function, the double angle identity involves the cosine double angle formula:
\[\sec(2\theta) = \frac{1}{\cos(2\theta)} = \frac{1}{2\cos^2(\theta) - 1}\]This formulation relies on the cosine double angle identity:
\[\cos(2\theta) = 2\cos^2(\theta) - 1\]By adapting the expression using these identities, you can solve and verify complex trigonometric expressions more easily.
Understanding double angle identities plays a crucial role when simplifying or validating trigonometric equations, such as the one you see in the exercise. You'll notice how each term in the identity directly relates to fundamental sine and cosine properties.

Trigonometric Simplification

Trigonometric simplification is the process of reducing complex trigonometric expressions to more manageable forms. This often involves substituting trigonometric identities or performing algebraic manipulations.
In the exercise, simplification starts by converting the expression on the right-hand side, \( \frac{\sec^2(\theta)}{2 - \sec^2(\theta)} \), into terms of \( \cos \theta \):

• Recall \( \sec^2(\theta) = \frac{1}{\cos^2(\theta)} \)
• Express the original as \( \frac{\frac{1}{\cos^2(\theta)}}{2 - \frac{1}{\cos^2(\theta)}} \)
• Simplify the denominator to \( \frac{2\cos^2(\theta) - 1}{\cos^2(\theta)} \)
• Perform division of fractions by multiplying by the reciprocal, simplifying to \( \frac{1}{2\cos^2(\theta) - 1} \)

The goal is to verify that the simplified form of both sides of a trigonometric identity is equal. With practice, these transformations become intuitive and essential for tackling various mathematical problems in trigonometry.