# Who is buried in Weierstrass' tomb?

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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Nice attention-grabbing title! :) –  anorton Jun 14 '13 at 15:53
I always refer to it as the universal trig substitution. –  Random Variable Jun 14 '13 at 16:00
I would guess that something like this would be known much earlier, by Euler and his contemporaries. –  Myself Jun 14 '13 at 16:21
I nominate Newton. –  Andreas Blass Jun 14 '13 at 16:41
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. –  MJD Jun 14 '13 at 17:10