# Weierstrass Substitution

## Proof Technique

The Weierstrass substitution is an application of Integration by Substitution.

The substitution is:

$u \leftrightarrow \tan \dfrac \theta 2$

for $-\pi < \theta < \pi$, $u \in \R$.

It yields:

 $\ds \sin \theta$ $=$ $\ds \frac {2 u} {1 + u^2}$ $\ds \cos \theta$ $=$ $\ds \frac {1 - u^2} {1 + u^2}$ $\ds \frac {\d \theta} {\d u}$ $=$ $\ds \frac 2 {1 + u^2}$

This can be stated:

$\ds \int \map F {\sin \theta, \cos \theta} \rd \theta = 2 \int \map F {\frac {2 u} {1 + u^2}, \frac {1 - u^2} {1 + u^2} } \frac {d u} {1 + u^2}$

where $u = \tan \dfrac \theta 2$.

## Proof

Let $u = \tan \dfrac \theta 2$ for $-\pi < \theta < \pi$.

From Shape of Tangent Function, this substitution is valid for all real $u$.

Then:

 $\ds u$ $=$ $\ds \tan \dfrac \theta 2$ $\ds \leadsto \ \$ $\ds \theta$ $=$ $\ds 2 \tan^{-1} u$ Definition of Inverse Tangent $\ds \leadsto \ \$ $\ds \dfrac {\d \theta} {\d u}$ $=$ $\ds \dfrac 2 {1 + u^2}$ Derivative of Arctangent Function and Derivative of Constant Multiple $\ds \sin \theta$ $=$ $\ds \dfrac {2 u} {1 + u^2}$ Tangent Half-Angle Substitution for Sine $\ds \cos \theta$ $=$ $\ds \dfrac {1 - u^2} {1 + u^2}$ Tangent Half-Angle Substitution for Cosine

The result follows from Integration by Substitution.

$\blacksquare$

## Also known as

This technique is also known as tangent half-angle substitution.

Some sources refer to these results merely as the half-angle formulas or half-angle formulae.

## Source of Name

This entry was named for Karl Theodor Wilhelm Weierstrass.