Mathematics

Mathematics

Problem

A student attempts to prove the identity (sec⁡x+tan⁡x)(sec⁡x−tan⁡x)=1(\sec x+\tan x)(\sec x-\tan x)=1 and writes the following work: Step 1: (sec⁡x+tan⁡x)(sec⁡x−tan⁡x)=sec⁡2x−tan⁡2x(\sec x+\tan x)(\sec x-\tan x)=\sec ^{2}x-\tan ^{2}x Step 2: sec⁡2x−tan⁡2x=sec⁡2x−(sec⁡2x−1)\sec ^{2}x-\tan ^{2}x=\sec ^{2}x-(\sec ^{2}x-1) Step 3: sec⁡2x−(sec⁡2x−1)=sec⁡2x−sec⁡2x+1=0\sec ^{2}x-(\sec ^{2}x-1)=\sec ^{2}x-\sec ^{2}x+1=0 Therefore (sec⁡x+tan⁡x)(sec⁡x−tan⁡x)=0(\sec x+\tan x)(\sec x-\tan x)=0. Answer the following: a. (i) Identify the step in which the logical or algebraic error first occurs. b. (ii) Rewrite the incorrect step correctly. c. (iii) Complete the proof of the identity (sec⁡x+tan⁡x)(sec⁡x−tan⁡x)=1(\sec x+\tan x)(\sec x-\tan x)=1 without the error.

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