The tangent half-angle substitution often used to anti-differentiate rational functions of sine and cosine, and also sometimes used to find closed-form solutions of some differential equations, is \begin{align} y & = \tan\frac x2 \\[8pt] \dfrac{1-y^2}{1+y^2} & = \cos x \\[8pt] \dfrac{2y}{1+y^2} & = \sin x \\[8pt] \dfrac{2\,dy}{1+y^2} & = dx \end{align}

Various books call this the Weierstrass substitution:

Is there historical evidence that this is due to Weierstrass, i.e. can it be found in something that he wrote?

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    $\begingroup$ Nice attention-grabbing title! :) $\endgroup$ – apnorton Jun 14 '13 at 15:53
  • $\begingroup$ I always refer to it as the universal trig substitution. $\endgroup$ – Random Variable Jun 14 '13 at 16:00
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    $\begingroup$ I would guess that something like this would be known much earlier, by Euler and his contemporaries. $\endgroup$ – Myself Jun 14 '13 at 16:21
  • $\begingroup$ I nominate Newton. $\endgroup$ – Andreas Blass Jun 14 '13 at 16:41
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    $\begingroup$ According to J-P. Merlet "Note on the History of Trigonometric Functions", published in International Symposium on History of Machines and Mechanisms, ed. Marco Caccarelli, Kluwer Academic Publishers, 2004, pp. 199: “All the authors seem to agree that this substitution was first used by Weierstrass (1815–1897).” But the only cite is to Stewart J. Single variable calculus. Brooks/Cole, 1994, which I would not consider authoritative. $\endgroup$ – MJD Jun 14 '13 at 17:10

Amazingly, if you dig down in the tomb of Weierstrass, you will find Euler! See

Euler, Institutiionum calculi integralis volumen primum, 1768, E342, Caput V, paragraph 261.

Go to http://www.eulerarchive.org/ and search for Index Number E342. There you will find the original Latin as well as an English translation by Ian Bruce.

This reference comes from Analysis by Its History by E. Hairer and G. Wanner (Springer 1991), p. 123. I have checked their reference and they are correct.

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    $\begingroup$ This reminds me of the rule for naming things Euler did: credit goes to the first person after him to invent something. en.wikipedia.org/wiki/List_of_things_named_after_Leonhard_Euler $\endgroup$ – Chris Janjigian Jun 15 '13 at 1:02
  • $\begingroup$ So this leaves the question of whether Weierstrass wrote anything about this, and how and why it got named after him. $\endgroup$ – Michael Hardy Jun 15 '13 at 18:33
  • $\begingroup$ This answer doesn't fully answer the question. It doesn't say whether Weierstrass wrote anything about this and it doesn't trace the history of the practice of naming it after him. The former question may be answerable only by going through everything Weierstrass wrote and failing to find this. The latter might or might not be easier. $\endgroup$ – Michael Hardy Jun 17 '13 at 12:35

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