Tangent half-angle substitution

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In integral calculus, the tangent half-angle substitution is a change of variables used for evaluating integrals, which converts a rational function of trigonometric functions of ${\textstyle x}$ into an ordinary rational function of ${\textstyle t}$ by setting ${\textstyle t=\tan {\tfrac {x}{2}}}$. This is the one-dimensional stereographic projection of the unit circle parametrized by angle measure onto the real line. The general^[1] transformation formula is:

$${\displaystyle \int f(\sin x,\cos x)\,dx=\int f{\left({\frac {2t}{1+t^{2}}},{\frac {1-t^{2}}{1+t^{2}}}\right)}{\frac {2\,dt}{1+t^{2}}}.}$$

The tangent of half an angle is important in spherical trigonometry and was sometimes known in the 17th century as the half tangent or semi-tangent.^[2] Leonhard Euler used it to evaluate the integral ${\textstyle \int dx/(a+b\cos x)}$ in his 1768 integral calculus textbook,^[3] and Adrien-Marie Legendre described the general method in 1817.^[4]

The substitution is described in most integral calculus textbooks since the late 19th century, usually without any special name.^[5] It is known in Russia as the universal trigonometric substitution,^[6] and also known by variant names such as half-tangent substitution or half-angle substitution. It is sometimes misattributed as the Weierstrass substitution.^[7] Michael Spivak called it the "world's sneakiest substitution".^[8]