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?
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.