Students were asked to prove the identity (tan x)(sin x) = sec x − cos x. Two students' work is given.
Student A
Step 1: sine x over cosine x times sine x equals secant x minus cosine x
Step 2: sine squared x over cosine x equals secant x minus cosine x
Step 3: 1 minus cosine squared x over cosine x equals secant x minus cosine x
Step 4: 1 over cosine x minus cosine squared x over cosine x equals secant x minus cosine x
Step 5: sec x − cos x = sec x − cos x
Student B
Step 1: tangent x times sine x equals 1 over cosine x minus cosine x
Step 2: tangent x times sine x equals 1 over cosine x minus cosine squared x over cosine x
Step 3: tangent x times sine x equals 1 minus cosine squared x over cosine x
Step 4: tangent x times sine x equals sine squared x over cosine x
Step 5: tangent x times sine x equals sine x over cosine x times sine x
Step 6: tan x sin x = tan x sin x
Part A: Did either student verify the identity properly? Explain why or why not. (10 points)
Part B: Name two identities that were used in Student A's verification and the steps they appear in. (5 points)
