Problem 99 - A Graphical Approach to Precalculus with Limits

Question

Verify that equation is an identity. \(\frac{\sec ^{4} s-\tan ^{4} s}{\sec ^{2} s+\tan ^{2} s}=\sec ^{2} s-\tan ^{2} s\)

Step-by-Step Solution

Answer

The given equation is an identity as both sides equal \( \sec^2 s - \tan^2 s \).

Step 1: Recall Trigonometric Identities

Use the identity \( \sec^2 s = 1 + \tan^2 s \) to help simplify the given expression.

Step 2: Simplify the Numerator

The numerator is \( \sec^4 s - \tan^4 s \). Recall that \( a^2 - b^2 = (a-b)(a+b) \). Here, \( a = \sec^2 s \) and \( b = \tan^2 s \), leading to \( (\sec^2 s - \tan^2 s)(\sec^2 s + \tan^2 s) \).

Step 3: Simplify the Whole Expression

Substitute the factorized numerator expression into the whole expression. This gives us \( \frac{(\sec^2 s - \tan^2 s)(\sec^2 s + \tan^2 s)}{\sec^2 s + \tan^2 s} \). Cancel out \( \sec^2 s + \tan^2 s \) from the numerator and the denominator.

Step 4: Verify the Identity

After canceling, we are left with \( \sec^2 s - \tan^2 s \), which is indeed the right side of the original equation, thus verifying it as an identity.

Key Concepts

Algebraic Manipulation

Algebraic manipulation involves rearranging expressions to make them simpler or to prove an identity. In this problem, we start with the given trigonometric expression and aim to verify its identity. We use the difference of squares formula:

• For two numbers or expressions, say \(a\) and \(b\), the formula is \(a^2 - b^2 = (a-b)(a+b)\).
• Now, in the context of our trigonometric expression, it's applied to break down the numerator \(\sec^4 s - \tan^4 s\).
• The expressions \(a\) and \(b\) are \(\sec^2 s\) and \(\tan^2 s\), respectively.
• This product form helps to reveal potential cancelling factors quickly, especially when paired with additional expressions like the denominator.

By leveraging these algebraic techniques, you can often transform complex expressions into simpler forms. This makes further manipulation and verifications not only possible but much more straightforward. Remember, success in algebraic manipulation often involves recognizing patterns and knowing which identities or formulas to apply.

Trigonometric Simplification

Trigonometric simplification uses well-known trigonometric identities to make an expression simpler or shorter. In this problem, a key starting point is the identity \(\sec^2 s = 1 + \tan^2 s\), one of the Pythagorean identities. Here's how simplification helps in our problem:

• Knowing that \(\sec^2 s = 1 + \tan^2 s\) allows us to replace complex expressions with simpler forms.
• When dealing with powers of \(\sec s\) and \(\tan s\), these identities help to consolidate terms.
• When the numerator \(\sec^4 s - \tan^4 s\) is rewritten as \((\sec^2 s - \tan^2 s)(\sec^2 s + \tan^2 s)\), it matches nicely with the denominator \(\sec^2 s + \tan^2 s\).

Simplifying trigonometric expressions is powerful because it helps us see our destination more clearly. The simplification leads directly to the answer, allowing features of the trigonometric functions to bring balance to both sides of the equation.

Verifying Identities

Verifying identities involves confirming that one side of an equation is equivalent to another. It's a common exercise in trigonometry that tests understanding of trigonometric functions and identities.

• Here, we start by assuming the left side is equal to the right side.
• Factorize, simplify, or manipulate one side using known identities like the Pythagorean identities or other algebraic techniques.
• After manipulation, both sides should clearly be identical in form and value.

For this example, once we've reduced the complex expression using algebraic manipulation and trigonometric simplification, we're left with \(\sec^2 s - \tan^2 s\) on both sides of the equation.This confirmation is crucial since the goal is to establish that an expression holds true universally for the variable involved.Understanding the logic and process of verification helps consolidate knowledge of identities, enabling easier handling of similar trigonometric problems.