Problem 43 - Precalculus: A Concise Course

Question

Use trigonometric identities to transform the left side of the equation into the right side \((0<\theta<\pi / 2)\). $$ (\sec \theta+\tan \theta)(\sec \theta-\tan \theta)=1 $$

Step-by-Step Solution

Answer

After applying the difference of squares formula and the Pythagorean identity, the left-hand side of the equation simplifies to 1, which matches the right-hand side. Thus, the transformation is verified.

Step 1: Apply the Difference of Squares Formula

We can regard \(\sec \theta\) as 'a' and \(\tan \theta\) as 'b', then apply the formula to the expression on the left-hand side. Therefore, we have that: \(\sec^2 \theta - \tan^2 \theta\).

Step 2: Use the Pythagorean Identity

Introduce the Pythagorean identity, which states that \(\sec^2\theta = 1 + \tan^2\theta\). This allows us to substitute \(\sec^2 \theta\) in our expression with \(1 + \tan^2\theta\). Now the equation becomes \(1 + \tan^2 \theta - \tan^2 \theta\).

Step 3: Simplify the Equation

Simplify the equation by eliminating the like-terms; \( \tan^2 \theta\) is subtracted from itself and completely removed from the equation. This leaves us with the equation equalling 1, which is the desired outcome: \(1 = 1\).

Key Concepts

Difference of Squares Formula

The Difference of Squares Formula is a foundational concept in algebra and calculus. It states that the product of the sum and difference of two terms can be expressed as the difference of their squares. In formula terms,

• \[(a+b)(a-b) = a^2 - b^2\]

This formula is extremely useful in simplifying expressions and solving equations.
In the context of trigonometry, it can help transform complex expressions into simpler forms by recognizing and utilizing such patterns.
In the problem given,

• The terms \[a = \sec \theta\] and \[b = \tan \theta\] are split into a difference of squares.

By recognizing the structure and applying this formula, the expression \((\sec \theta + \tan \theta)(\sec \theta - \tan \theta)\) becomes \[\sec^2 \theta - \tan^2 \theta\].This transformation is the first step toward simplifying the equation.

Pythagorean Identity

Pythagorean Identities are essential tools in trigonometry that relate the squares of sine, cosine, and tangent functions to 1. These identities derive from the Pythagorean Theorem, common in geometry.
Specifically, the Pythagorean Identity that we use here is:

• \[\sec^2 \theta = 1 + \tan^2 \theta\]

This is crucial because it provides a way to substitute one function's squared value with terms more conducive to simplification. In this problem, replacing \[\sec^2 \theta\] with \[1 + \tan^2 \theta\] transforms a potentially complex equation into something much simpler.
After applying this identity, the expression becomes \[1 + \tan^2 \theta - \tan^2 \theta\].
This step simplifies the equation significantly by setting the stage for final cancellation of terms.

Angle Simplification

Angle Simplification involves reducing an expression to its most simple form, often by eliminating unnecessary components or combining like-terms. As a final step in the solution process for trigonometric identities, simplification confirms equivalence or solves equations.
Once we get to \[1 + \tan^2 \theta - \tan^2 \theta\],
we notice that the \[\tan^2 \theta\] terms cancel each other out.

• This leaves us with \[1 = 1\], confirming the identity transformation.

This not only demonstrates our initial equation's truth but also highlights how breaking down and methodically simplifying can lead to clarity and understanding.
This skill is invaluable for students as they work through complex trigonometric equations.