Problem 97 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

The equation is an identity.

Step 1: Express in terms of basic identities

We know the identities: 1. \( \sec^2 x = 1 + \tan^2 x \)2. Therefore, \( \sec^4 x = (\sec^2 x)^2 = (1 + \tan^2 x)^2 \).Substituting back into the left-hand side of the equation, we have: \[ \sec^4 x - \sec^2 x = (1 + \tan^2 x)^2 - (1 + \tan^2 x) \].

Step 2: Expand the identity

Expand \((1 + \tan^2 x)^2:\)\[ (1 + \tan^2 x)(1 + \tan^2 x) = 1 + 2\tan^2 x + \tan^4 x \].Substitute back into the equation: \[ \sec^4 x - \sec^2 x = 1 + 2\tan^2 x + \tan^4 x - (1 + \tan^2 x) \].

Step 3: Simplify the expression

Simplify the expression by subtracting:\[ 1 + 2\tan^2 x + \tan^4 x - 1 - \tan^2 x = \tan^4 x + \tan^2 x \].This simplifies to the right-hand side of our original equation, \( \tan^4 x + \tan^2 x \).

Step 4: Conclude the identity verification

Both sides are equivalent:The original equation \( \sec^4 x - \sec^2 x = \tan^4 x + \tan^2 x \) has been transformed and simplified to show equality between the left-hand side and the right-hand side, verifying that it is indeed an identity.

Key Concepts

Secant and Tangent Identities

In trigonometry, certain functions are related through identities, which are equations that are true for all values within an interval.
For instance, the secant (\( \sec \, x \)) and tangent (\( \tan \, x \)) are hyperbolic functions with their own set of identities.
One of the main identities for secant and tangent is:

• \( \sec^2 x = 1 + \tan^2 x \)

This identity is an essential building block for many other trigonometric equations.
To extend this, we can square both sides of the identity to find a more complex relation:

• \( \sec^4 x = (1 + \tan^2 x)^2 \)

These identities help us rewrite complex equations in simpler forms,
facilitating the process of solving trigonometric equations and verifying identities.

Algebraic Manipulation

Algebraic manipulation involves using algebraic rules to transform expressions.
This includes expanding and simplifying expressions, which are common tools in verifying trigonometric identities.
In the given exercise, we start by expressing \( \sec^4 x \) in terms of tangent using the identity:

• \( \sec^4 x = (1 + \tan^2 x)^2 \)

Then, we expand the squared term:

• \((1 + \tan^2 x)(1 + \tan^2 x) = 1 + 2\tan^2 x + \tan^4 x \)

Next, we combine this expanded form with the original equation,
and proceed to subtract:

• \(1 + 2\tan^2 x + \tan^4 x - 1 - \tan^2 x \)

Algebraic manipulation simplifications lead us to observe that many terms cancel out,
eventually leaving us with a simplified form: \( \tan^4 x + \tan^2 x \).
These steps confirm that algebraic manipulation is crucial for simplifying and verifying identities.

Identity Verification

Identity verification involves showing that both sides of an equation are equivalent for all values of the variable.
In this exercise, our goal is to verify \( \sec^4 x - \sec^2 x = \tan^4 x + \tan^2 x \).
Initially, the left-hand side is transformed using trigonometric identities and algebraic manipulation.By substituting the expanded and simplified form, we reach the equation:

• \( \tan^4 x + \tan^2 x \)

which matches the right-hand side
This demonstrates that both sides are indeed equal for all values of \( x \) for which the identity is defined.
Completing a verification confirms the truth of the identity through logical and mathematical processes. This results in a reliable equation that holds true universally,
showcasing the effectiveness of using identities and manipulation in trigonometry.