1. The problem statement, all variables and given/known data Show that the two formulas are equivalent integral [sec x dx] = ln|sec x + tan x| + C integral [sec x dx] = -ln|sec x - tan x| + C 2. Relevant equations Pythagorean ID's? Log rule of addition 3. The attempt at a solution Well, I realized the formulas can only be equivalent if -ln|sec x - tan x| + C = ln|sec x + tan x| + C I put the constants on one side, ln on the other: ln |sec x + tan x| + ln| sec x - tan x| = C Then, ln |(sec x + tan x) * (sec x - tan x)| = C ln |(sec x)^2 - (tan x)^2| = C Using pythagorean ID's, ln |1| = C But what now, 0 = C doesn't show the two identities are equivalent... am I also supposed to assume in -ln|sec x - tan x| + C = ln|sec x + tan x| + C that the constants are equivalent so that ln |sec x + tan x| + ln| sec x - tan x| = 0 ??? Because then ln |1| = 0 ?? I thought that constants must be different and that C-C doesn't necessarily = 0 so you can't assume that.. Did I approach this whole thing wrong? Even if ln |1| = 0, how would that prove that -ln|sec x - tan x| + C = ln|sec x + tan x| + C it simplifies to ln |1| = 0, but just because the simplified result is a true statement (possibly not because what if ln |1| = C does that prove the original equations are equivalent? or .....?