Problem 74 - Contemporary Precalculus

Question

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

Step-by-Step Solution

Answer

Question: Prove the identity: $$\frac{\cos^3{x} - \sin^3{x}}{\cos{x} - \sin{x}} = 1 + \sin{x}\cos{x}$$ Answer: Following the steps outlined in the solution, we can simplify and prove the given identity as: $$\frac{\cos^3{x} - \sin^3{x}}{\cos{x} - \sin{x}} = \cos^2{x} + \cos{x}\sin{x} + \sin^2{x} = 1 + \sin{x}\cos{x}$$

Step 1: Factor the numerator using the difference of cubes formula

We'll start by factoring the numerator of the left side using the difference of cubes formula (a^3 - b^3 = (a - b)(a^2 + ab + b^2)). In this case, a = cos x, and b = sin x. $$ \frac{\cos^3{x} - \sin^3{x}}{\cos{x} - \sin{x}} = \frac{(\cos{x} - \sin{x})(\cos^2{x} + \cos{x}\sin{x} + \sin^2{x})}{\cos{x} - \sin{x}} $$

Step 2: Cancel out the common terms

Notice that the term \((\cos{x} - \sin{x})\) is common in both the numerator and denominator; therefore, we can cancel this term out: $$ \frac{(\cos{x} - \sin{x})(\cos^2{x} + \cos{x}\sin{x} + \sin^2{x})}{\cos{x} - \sin{x}} = \cos^2{x} + \cos{x}\sin{x} + \sin^2{x} $$

Step 3: Use the Pythagorean trigonometric identity

We know that \(\cos^2{x} + \sin^2{x} = 1\), which is a Pythagorean trigonometric identity. Substitute this identity into our expression from Step 2: $$ \cos^2{x} + \cos{x}\sin{x} + \sin^2{x} = 1 + \cos{x}\sin{x} $$ We have now shown that the given expression simplifies to the right side of the equation. Therefore, the identity is proved: $$ \frac{\cos^3{x} - \sin^3{x}}{\cos{x} - \sin{x}} = 1 + \sin{x}\cos{x} $$

Key Concepts

Difference of Cubes

In mathematics, factoring is a crucial skill that simplifies more complex expressions into manageable components. The difference of cubes formula is a specific type of factoring. It states that for any two terms, say \(a\) and \(b\), the expression \(a^3 - b^3\) can be rewritten as \((a - b)(a^2 + ab + b^2)\).
\[ A^3 - B^3 = (A - B)(A^2 + AB + B^2) \]
This formula is useful in simplifying expressions that may initially seem daunting. In the exercise, the expression \(\cos^3{x} - \sin^3{x}\) is factored using this formula by assigning \(a = \cos{x}\) and \(b = \sin{x}\).

• Evaluate the expression as each cube involves cosine and sine of \(x\).
• Apply the difference of cubes formula as a useful tool.

By factoring this way, it becomes easier to identify and cancel the common terms with expressions in the denominator. This step is fundamental before proceeding to any further simplifications.

Pythagorean Identity

The Pythagorean Identity is a fundamental trigonometric identity that relates the squares of sine and cosine functions. It is written as:\[\cos^2{x} + \sin^2{x} = 1\]This identity emerges from the Pythagorean theorem and is particularly handy in simplifying trigonometric expressions by replacing complex sums of squares with the number 1. In our exercise, after factoring and canceling terms using the difference of cubes formula, the expression involved \(\cos^2{x} + \sin^2{x}\).

Here’s how we applied it:

• The term \(\cos^2{x} + \sin^2{x}\) directly simplifies to 1.
• This transformation enables the expression to align with the right side of the given identity \(1 + \sin{x}\cos{x}\).

Utilizing this identity effectively transforms the expression and provides a direct path to proving the identity. It simplifies calculations and helps in focusing on the trigonometric function values rather than handling complex algebraic expressions.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions involves the art of using algebraic techniques alongside known trigonometric identities to rewrite expressions in a simpler or more revealing form. For this exercise, the goal was to transform a complex expression into a simpler form which can ultimately show that it equals another expression.

Here's a general approach:

• Identify patterns or identities in the expression that can be simplified, such as factoring or using trigonometric identities like the Pythagorean Identity.
• Cancel common factors to reduce complexity, such as removing identical terms in the numerator and denominator.
• Substitute known identities to replace complex terms with simpler equivalents.

These techniques were employed to streamline the expression from the complex fraction \(\frac{\cos^3{x} - \sin^3{x}}{\cos{x} - \sin{x}}\) to the much simpler \(1 + \sin{x}\cos{x}\). Each step in simplifying such expressions not only reduces the work involved but also illuminates the beauty and interconnectedness of trigonometric identities.