Undoing Trig Substitution Professor Miller plays a game in which students give him a trig function and an inverse trig function, and then he tries to compute their composition. As we’ve seen, this is sometimes the ﬁnal step in integration by trig substitution. (arc x) =? Example: tan(arccsc x) =? Question: Isn’t tan(arccsc x) acceptable as a ﬁnal answer? Answer: What does “acceptable” mean? The expression − csc(arctan x) √ 1 + x2 was a correct ﬁnal answer, but is a nicer, more insightful, and probably x more useful answer. To simplify tan(arccsc x) we draw a triangle illustrating an angle whose cosecant is x; see Figure 1. We know that x = csc θ = 1 hyp = sin θ opp so we choose convenient values x and 1 to be the lengths of the hypotenuse and opposite side. x θ √ 1 x2 − 1 Figure 1: θ = arccsc x so x = csc θ. Once we’ve drawn √ our triangle we can compute that the length of the adja cent side must be x2 − 1, and so tan θ = opp 1 =√ . 2 adj x −1 Since x = csc θ, we have: 1 tan(arccsc x) = tan θ = √ . 2 x −1 Whenever you have to undo a trig substitution, this technique is likely to be useful.