Problem 30 - Precalculus: Functions and Graphs

Question

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

Step-by-Step Solution

Answer

The identity is verified: \( \frac{\cos^3 x - \sin^3 x}{\cos x - \sin x} = 1 + \sin x \cos x \).

Step 1: Recognize the Formula for a Difference of Cubes

Notice that the numerator \( \cos^3 x - \sin^3 x \) can be rewritten using the difference of cubes formula, which is \( a^3 - b^3 = (a-b)(a^2 + ab + b^2) \). Let \( a = \cos x \) and \( b = \sin x \). Then, \( \cos^3 x - \sin^3 x = (\cos x - \sin x)(\cos^2 x + \cos x \sin x + \sin^2 x) \).

Step 2: Simplify the Expression

The expression \( \cos^2 x + \cos x \sin x + \sin^2 x \) simplifies because we know that \( \cos^2 x + \sin^2 x = 1 \). Thus, \( \cos^2 x + \sin^2 x + \cos x \sin x = 1 + \cos x \sin x \).

Step 3: Substitute Back into the Identity

Substitute the simplified expression back into the fraction: \[\frac{(\cos x - \sin x)(1 + \cos x \sin x)}{\cos x - \sin x}.\]The \( \cos x - \sin x \) terms in the numerator and denominator cancel each other out, leaving \( 1 + \cos x \sin x \).

Step 4: Verify the Identity

After cancellation, the expression simplifies to \( 1 + \cos x \sin x \), which matches the right-hand side of the original identity, confirming that the identity is valid.

Key Concepts

Difference of Cubes

When we encounter expressions like \( \cos^3 x - \sin^3 x \), recognizing them as a "difference of cubes" can be an efficient way to simplify them. The difference of cubes formula is:

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

In this formula, \( a \) and \( b \) are placeholders for any terms. Applying this to our expression, we let \( a = \cos x \) and \( b = \sin x \). Substitute these into the formula:

• \( \cos^3 x - \sin^3 x = (\cos x - \sin x)(\cos^2 x + \cos x \sin x + \sin^2 x) \)

Using the difference of cubes helps break down complex algebraic expressions, making them simpler to handle. This technique is useful because it allows us to factor expressions neatly and often leads to further simplification.

Simplifying Expressions

Simplifying expressions is all about reducing them to their most concise representation. For a trigonometric identity, recognizing fundamental truths or stepping stones can greatly help. Take this expression: \( \cos^2 x + \cos x \sin x + \sin^2 x \).
Using the Pythagorean identity, \( \cos^2 x + \sin^2 x = 1 \), makes this process smoother.

• Thus, \( \cos^2 x + \sin^2 x + \cos x \sin x \) simplifies to \( 1 + \cos x \sin x \).

By simplifying expressions strategically, we decrease the complexity we have to work with. Here, knowing basic identities like the Pythagorean identity can turn a longer expression into something quite compact and simple. This process often involves canceling out terms, factoring where possible, and substituting known identities.

Trigonometric Simplification

Trigonometric simplification is an invaluable tool when verifying identities, simplifying expressions, or solving trigonometric equations. It's key to always remind yourself of the basic trigonometric identities. Checking the original identity \( \frac{\cos^3 x - \sin^3 x}{\cos x - \sin x}=1+\sin x \cos x \) illustrates the power of trigonometric simplification.
After applying the difference of cubes formula, simplification further led us to cancel the common terms in the equation. This process:

• Started by rewriting the numerator \( \cos^3 x - \sin^3 x \) as \( (\cos x - \sin x)(1 + \cos x \sin x) \).
• The common factor \( \cos x - \sin x \) was then canceled, simplifying the expression to \( 1 + \cos x \sin x \).

With practice, these simplifications turn a seemingly difficult problem into a straightforward check. Trigonometric simplification often involves finding commonalities, using identities, or rewriting expressions in more manageable terms. It's the smart way of solving seemingly intricate trigonometric puzzles.