Question

Students were asked to prove the identity (sec x)(csc x) = cot x + tan x. Two students' work is given.Student AStep 1: 1 over cosine x times 1 over sine x equals cotangent x plus tangent xStep 2: 1 over cosine x times sine x equals cotangent x plus tangent xStep 3: cosine squared x plus sine squared x over cosine x times sine x equals cotangent x plus tangent xStep 4: cosine squared x over cosine x times sine x plus sine squared x over cosine x times sine x equals cotangent x plus tangent xStep 5: cosine x over sine x plus sine x over cosine x equals cotangent x plus tangent xStep 6: cot x + tan x = cot x + tan xStudent BStep 1: secant x times cosecant x equals cosine x over sine x plus sine x over cosine xStep 2: secant x times cosecant x equals cosine squared x over cosine x times sine x plus sine squared x over cosine x times sine xStep 3: secant x times cosecant x equals cosine squared x over cosine x times sine x plus sine squared x over cosine x times sine xStep 4: secant x times cosecant x equals 1 over cosine x times sine xStep 5: secant x times cosecant x equals 1 over cosine x times 1 over sine xStep 6: sec x csc x = sec x csc xPart 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)

Answer 1

Given:

Part A:

`\begin{gathered} student\text{ A verified the identity properly} \\ Reason:\text{ student A applied the trigonometric identities} \end{gathered}`

Part B:

The identities used in student A verification are

`\begin{gathered} step\text{ 1: sec x = }\frac{1}{cos\text{ x}};\text{ csc x =}\frac{1}{sin\text{ x}} \\ step\text{ 3: }\cos^2x+\sin^2x=1 \\ step\text{ 5: }(\cos x)/(\sin x)=cot\text{ x ; }\frac{\sin x}{\cos\text{ x}}=\tan\text{ x} \\ \\ \end{gathered}`