Tangent half-angle substitution

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 transformation formula is:

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${\displaystyle f(a)-f(b)=\int _{b}^{a}f'(t)\,dt}$
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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. Leonhard Euler used it to evaluate the integral ${\textstyle \int dx/(a+b\cos x)}$ in his 1768 integral calculus textbook, and Adrien-Marie Legendre described the general method in 1817.

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