Problem 85 - Precalculus Mathematics for Calculus

Question

Verify the identity. $$\frac{\sin ^{3} x+\cos ^{3} x}{\sin x+\cos x}=1-\sin x \cos x$$

Step-by-Step Solution

Answer

The identity is verified: both sides equal \( 1 - \sin x \cos x \).

Step 1: Define Identity and Simplify Left Side

We are given the identity \( \frac{\sin^3 x + \cos^3 x}{\sin x + \cos x} = 1 - \sin x \cos x \). We first simplify the left side using the sum of cubes formula: \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \). Here, \( a = \sin x \) and \( b = \cos x \). Therefore, the numerator becomes \( (\sin x + \cos x)(\sin^2 x - \sin x \cos x + \cos^2 x) \).

Step 2: Cancel Out Common Terms

The denominator \( \sin x + \cos x \) cancels out with the factor in the numerator \( \sin x + \cos x \), leaving us with \( \sin^2 x - \sin x \cos x + \cos^2 x \).

Step 3: Use Pythagorean Identity

Recall the Pythagorean identity: \( \sin^2 x + \cos^2 x = 1 \). Replace \( \sin^2 x + \cos^2 x \) in the expression \( \sin^2 x - \sin x \cos x + \cos^2 x \) with 1, leaving us with \( 1 - \sin x \cos x \).

Step 4: Compare Both Sides of the Equation

The expression obtained \( 1 - \sin x \cos x \) matches the right side of the identity. This verifies that the identity is correct.

Key Concepts

Sum of Cubes Formula

The sum of cubes formula is a handy algebraic identity used to simplify expressions involving cubed terms. This formula is expressed as:

• \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \)

It allows us to break down complex expressions involving cubes into a product of a binomial and a trinomial. This decomposition makes further algebraic manipulations much easier.

In the given exercise, \( a \) is equated to \( \sin x \) and \( b \) to \( \cos x \). Using the sum of cubes formula, the expression \( \sin^3 x + \cos^3 x \) can be rewritten as:

• \( (\sin x + \cos x)(\sin^2 x - \sin x \cos x + \cos^2 x) \)

This step is crucial because it brings a factor \( \sin x + \cos x \) in both the numerator and denominator, allowing us to simplify the fraction significantly.

Pythagorean Identity

The Pythagorean identity is a fundamental concept in trigonometry, connecting sine and cosine functions. This identity states:

• \( \sin^2 x + \cos^2 x = 1 \)

This relationship is derived from the Pythagorean theorem applied to a unit circle and is used frequently to simplify trigonometric expressions.

In our exercise, after reducing the initial expression using the sum of cubes, we are left with \( \sin^2 x - \sin x \cos x + \cos^2 x \). By recognizing the presence of \( \sin^2 x + \cos^2 x \) in this expression, we can substitute it with 1, according to the Pythagorean identity. This simplifies the expression to \( 1 - \sin x \cos x \), directly addressing the task of verifying the given identity.

Simplifying Expressions

Simplifying expressions is a critical skill in mathematics that involves reducing them to their simplest or most manageable form. By simplifying complex trigonometric expressions, one can make equations much easier to handle and solve.

• Identify common factors and cancel them out.
• Use known identities, such as the Pythagorean identity, to substitute equivalent expressions.
• Apply algebraic identities, like the sum of cubes, for breaking down challenging terms.

In the original exercise, we simplified a complex trigonometric expression step-by-step. By canceling common terms in the fraction and leveraging identities effectively, we ended with a concise expression: \( 1 - \sin x \cos x \).

The outcome demonstrates how strategic simplification not only verifies identities but also enhances understanding, making seemingly complicated expressions more approachable and solvable.