Problem 109 - A Graphical Approach to Precalculus with Limits

Question

Verify that each equation is an identity by using any of the identities introduced in the first three sections of this chapter. $$\frac{\cos ^{4} x-\sin ^{4} x}{\cos ^{2} x}=1-\tan ^{2} x$$

Step-by-Step Solution

Answer

The identity is valid as both sides simplify to \( 1 - \tan^2 x \).

Step 1: Recognize a Difference of Squares

The numerator of the left-hand side of the equation, \( \cos^4 x - \sin^4 x \), can be rewritten using the difference of squares formula \( a^2 - b^2 = (a-b)(a+b) \). In this case, set \( a = \cos^2 x \) and \( b = \sin^2 x \). Hence, \( \cos^4 x - \sin^4 x = (\cos^2 x - \sin^2 x)(\cos^2 x + \sin^2 x) \).

Step 2: Simplify Using the Pythagorean Identity

Now, apply the Pythagorean identity \( \cos^2 x + \sin^2 x = 1 \) to simplify the expression. Therefore, \((\cos^2 x - \sin^2 x)(\cos^2 x + \sin^2 x) = (\cos^2 x - \sin^2 x) \) because \( \cos^2 x + \sin^2 x = 1 \).

Step 3: Substitute into the Original Equation

Substitute the simplified expression for \(\cos^4 x - \sin^4 x\) into the original identity: \(\frac{\cos^2 x - \sin^2 x}{\cos^2 x} \).

Step 4: Further Simplify

Simplify \( \frac{\cos^2 x - \sin^2 x}{\cos^2 x} \) by separating the fraction: \( \frac{\cos^2 x}{\cos^2 x} - \frac{\sin^2 x}{\cos^2 x} = 1 - \tan^2 x \), where \( \tan^2 x = \frac{\sin^2 x}{\cos^2 x} \).

Step 5: Verify Both Sides Equal

Now, observe that the left-hand side simplifies to exactly \( 1 - \tan^2 x \), which matches the right-hand side of the original identity, confirming the identity is valid.

Key Concepts

Difference of Squares

Understanding the difference of squares is essential in algebra. It is a simple yet powerful formula: \(a^2 - b^2 = (a-b)(a+b)\). This means any difference between two squares can be factored into a product of two binomials. In the exercise, \(\cos^4 x - \sin^4 x\) is expressed as a difference of squares. We identify \(a = \cos^2 x\) and \(b = \sin^2 x\), allowing us to rewrite the expression as \((\cos^2 x - \sin^2 x)(\cos^2 x + \sin^2 x)\). Factoring this way makes the expression much simpler to work with. Recognizing such patterns is key in solving complex algebraic identities and can serve as a helpful tool in trigonometry.

Pythagorean Identity

The Pythagorean identity is a fundamental trigonometric identity that states \(\cos^2 x + \sin^2 x = 1\). It is derived from the Pythagorean theorem and is used extensively across various branches of math and physics. In our exercise, after applying the difference of squares, we can replace \(\cos^2 x + \sin^2 x\) with 1. This significantly simplifies the expression. By substituting 1 for \(\cos^2 x + \sin^2 x\), the original complex-looking fraction \((\cos^2 x - \sin^2 x)(\cos^2 x + \sin^2 x)\) reduces to \(\cos^2 x - \sin^2 x\). This identity simplifies expressions and helps easily transition between different forms of trigonometric equations. Memorizing and understanding it is crucial for solving trigonometric identities and proofs.

Simplification Techniques

Simplification techniques in mathematics are methods used to make expressions easier to work with by reducing their complexity. Breaking down expressions allows us to see connections and relationships more clearly.In this exercise, once we applied the difference of squares and the Pythagorean identity, we simplified the fraction: \(\frac{\cos^2 x - \sin^2 x}{\cos^2 x}\). By dividing each term of the numerator by the denominator separately, we break it down into simpler fractions: \(\frac{\cos^2 x}{\cos^2 x} - \frac{\sin^2 x}{\cos^2 x}\).

• The term \(\frac{\cos^2 x}{\cos^2 x}\) simplifies to 1.
• The term \(\frac{\sin^2 x}{\cos^2 x}\) simplifies to \(\tan^2 x\), because \(\tan x = \frac{\sin x}{\cos x}\).

Combining these results gives us \(1 - \tan^2 x\), exactly matching the right side of the equation. Simplification not only confirms the identity but also makes intricate equations more manageable.