Problem 78 - Algebra and Trigonometry

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

0 out of 10 points should be awarded because the student's method of solving the identity is mathematically incorrect.

Step 1: Identifying the Problem

First, let's examine the student's solution. The student has split the fraction \(\frac{\sin^{2}x - \cos^{2}x}{\sin x + \cos x}\) into \(\frac{\sin^{2}x}{\sin x} - \frac{\cos^{2}x}{\cos x}\) which reduces to \(\sin x - \cos x\).

Step 2: Evaluating the Student's Solution

This is not a valid method to prove the identity because dividing the terms of the numerator and the terms of the denominator doesn't hold true in general unless both terms of the denominator divides all the terms of the numerator, which is not the case here.

Step 3: Assigning Points

As the method used by the student is mathematically incorrect, no points should be given for incorrect process.