Problem 78 - Precalculus Essentials

Question

A 10 -point question on a quiz asks students to verify the identity $$\frac{\sin ^{2} x-\cos ^{2} x}{\sin x+\cos x}=\sin x-\cos x$$ One student begins with the left side and obtains the right side as follows: $$\frac{\sin ^{2} x-\cos ^{2} x}{\sin x+\cos x}=\frac{\sin ^{2} x}{\sin x}-\frac{\cos ^{2} x}{\cos x}=\sin x-\cos x$$ How many points (out of 10 ) would you give this student? Explain your answer.

Step-by-Step Solution

Answer

The student made a common math mistake and doesn't truly understand the identities involved, so perhaps 2-3 points out of 10 would be a fair score.

Step 1: Identity of the Misstep

The student started with \(\frac{\sin^2(x)-\cos^2(x)}{\sin(x)+\cos(x)}\) and rewrote it incorrectly as \(\frac{\sin^2(x)}{\sin(x)} - \frac{\cos^2(x)}{\cos(x)}\). What the student did wrong here is a common error in algebra often referred to as 'false factorisation'. When subtracting or adding fractions, it is not possible to divide the numerator and the denominator separately as the student has done in their work.

Step 2: Correct Approach

The correct approach to proving this trigonometric identity starts with the left side of the given equation: \(\frac{\sin^2(x)-\cos^2(x)}{\sin(x)+\cos(x)}\). This is equivalent to \(\frac{(\sin(x)+\cos(x))(\sin(x)-\cos(x))}{\sin(x) + \cos(x)}\). Cancelling the \(\sin(x) + \cos(x)\) on the top and bottom of the fraction, this simplifies to \(\sin(x)-\cos(x)\) which is the right side of the initial equation.

Step 3: Scoring the Student's Solution

The student's method was flawed, and they ended up at the correct answer without truly validating the identity. Therefore, it can be considered that the student does not fully understand the problem or the math involved. Then again, the student did recognize that this is indeed a problem involving trigonometric identities, so some comprehension is present. Consequently, a fair score might be 2 to 3 points out of 10.

Key Concepts

Understanding Algebra Errors

Algebra errors often stem from misunderstandings of basic algebraic principles. These mistakes can lead to incorrect results in mathematical problems, especially when simplifying expressions or converting equations. One common misconception involves the notion that terms in a numerator and denominator can be separated and treated independently, as seen when they incorrectly apply distribution laws or factorization.

• This results in the separate treatment of terms in fractions, which should never happen.
• Algebra demands that operations adhere strictly to operational rules, meaning that both the numerator and denominator must be considered as whole entities when performing arithmetic operations.

Misunderstanding this concept can lead to answers that seem correct but are not derived through legitimate processes. Recognizing and rectifying such errors strengthens one's foundational mathematics skills, ensuring better accuracy in solving problems. To avoid these mistakes, always reassess whether the applied operations align with algebraic laws, ensuring the entire mathematical structure remains intact.

The Pitfalls of False Factorization

False factorization is an error where an expression is factored incorrectly. This error often leads to incorrect simplifications and, consequently, erroneous solutions. In trigonometry and algebra, attention to detail is crucial.

• False factorization occurs when assumptions about common factors are incorrectly made or applied without valid operations.
• It's crucial to remember that each component of an expression contributes to the whole, and incorrect factoring disrupts this integral connection.

To illustrate, consider an expression such as \((a^2 - b^2)\) which correctly factors to \((a+b)(a-b)\). False factorization might lead one to separate terms incorrectly, assuming an inaccurate simplification of the form \(\frac{a^2}{a} - \frac{b^2}{b}\), which isn't valid without mutual factors across terms.This type of error typically arises from haste or misunderstanding factorization rules. Avoiding such errors requires a solid understanding of algebraic identities and controlled, step-by-step problem-solving approaches.

The Process of Verifying Identities

Verifying identities in trigonometry involves confirming that two sides of an equation are equal under all conditions. This demands a precise approach where alterations to one side of an identity transform it into the other without violating mathematical laws.

• Verifying identities involves simplifying or transforming one side of an equation to match the other.
• Cleaving to algebraic and trigonometric identities and transformation methods is essential.
• These include strategies like factoring, distributing, and using known identities like the Pythagorean identity.

Proper verification necessitates careful substitution and simplification, abiding by established mathematical axioms. Errors often arise from shortcuts or misapplications of rules, leading to conclusions that might seem correct but lack rigorous backing. For students learning to verify identities, the key lies in mastering algebraic principles and the correct application of trigonometric identities, ensuring that the process is both logical and systematic. By doing so, each step aligns faithfully with mathematical truths, leading to authentic and verifiable results.