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Concept
Tangent half-angle substitution
Summary
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 into an ordinary rational function of by setting . This is the one-dimensional stereographic projection of the unit circle parametrized by angle measure onto the real line. The general transformation formula is:
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 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.James Stewart mentioned Karl Weierstrass when discussing the substitution in his popular calculus textbook, first published in 1987:
Later authors, citing Stewart, have sometimes referred to this as the Weierstrass substitution, for instance:
Stewart provided no evidence for the attribution to Weierstrass. A related substitution appears in Weierstrass’s Mathematical Works, from an 1875 lecture wherein Weierstrass credits Carl Gauss (1818) with the idea of solving an integral of the form by the substitution
Michael Spivak called it the "world's sneakiest substitution".
Introducing a new variable sines and cosines can be expressed as rational functions of and can be expressed as the product of and a rational function of as follows:
Using the double-angle formulas, introducing denominators equal to one thanks to the Pythagorean theorem, and then dividing numerators and denominators by one gets
Finally, since , differentiation rules imply
and thus
We can confirm the above result using a standard method of evaluating the cosecant integral by multiplying the numerator and denominator by and performing the substitution .
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