# Weierstrass substitution formulas

The Weierstrass substitution formulas for $-\pi are:

$\begin{array}[]{rl}\sin x&=\displaystyle\frac{2t}{1+t^{2}}\\ &\\ \cos x&=\displaystyle\frac{1-t^{2}}{1+t^{2}}\\ &\\ dx&=\displaystyle\frac{2}{1+t^{2}}\,dt\end{array}$

They can be obtained in the following manner:

Make the Weierstrass substitution $\displaystyle t=\tan\left(\frac{x}{2}\right)$. (This substitution is also known as the universal trigonometric substitution.) Then we have

$\begin{array}[]{rl}\displaystyle\cos\left(\frac{x}{2}\right)&=\displaystyle% \frac{1}{\displaystyle\sec\left(\frac{x}{2}\right)}\\ &\\ &=\displaystyle\frac{1}{\displaystyle\sqrt{1+\tan^{2}\left(\frac{x}{2}\right)}% }\\ &\\ &=\displaystyle\frac{1}{\sqrt{1+t^{2}}}\end{array}$

and

$\begin{array}[]{rl}\displaystyle\sin\left(\frac{x}{2}\right)&=\displaystyle% \cos\left(\frac{x}{2}\right)\cdot\tan\left(\frac{x}{2}\right)\\ &\\ &=\displaystyle\frac{t}{\sqrt{1+t^{2}}}.\end{array}$

Note that these are just the “formulas involving radicals (http://planetmath.org/Radical6)” as designated in the entry goniometric formulas; however, due to the restriction on $x$, the $\pm$’s are unnecessary.

Using the above formulas along with the double angle formulas, we obtain

$\begin{array}[]{rl}\sin x&=\displaystyle 2\sin\left(\frac{x}{2}\right)\cdot% \cos\left(\frac{x}{2}\right)\\ &\\ &=\displaystyle 2\cdot\frac{t}{\sqrt{1+t^{2}}}\cdot\frac{1}{\sqrt{1+t^{2}}}\\ &\\ &=\displaystyle\frac{2t}{1+t^{2}}\end{array}$

and

$\begin{array}[]{rl}\cos x&=\displaystyle\cos^{2}\left(\frac{x}{2}\right)-\sin^% {2}\left(\frac{x}{2}\right)\\ &\\ &=\displaystyle\left(\frac{1}{\sqrt{1+t^{2}}}\right)^{2}-\left(\frac{t}{\sqrt{% 1+t^{2}}}\right)^{2}\\ &\\ &=\displaystyle\frac{1}{1+t^{2}}-\frac{t^{2}}{1+t^{2}}\\ &\\ &=\displaystyle\frac{1-t^{2}}{1+t^{2}}.\end{array}$

Finally, since $\displaystyle t=\tan\left(\frac{x}{2}\right)$, solving for $x$ yields that $x=2\arctan t$. Thus, $\displaystyle dx=\frac{2}{1+t^{2}}\,dt$.

The Weierstrass substitution formulas are most useful for integrating rational functions of sine and cosine (http://planetmath.org/IntegrationOfRationalFunctionOfSineAndCosine).

 Title Weierstrass substitution formulas Canonical name WeierstrassSubstitutionFormulas Date of creation 2013-03-22 17:05:25 Last modified on 2013-03-22 17:05:25 Owner Wkbj79 (1863) Last modified by Wkbj79 (1863) Numerical id 12 Author Wkbj79 (1863) Entry type Definition Classification msc 26A36 Classification msc 33B10 Synonym Weierstraß substitution formulas Related topic GoniometricFormulae Related topic IntegrationOfRationalFunctionOfSineAndCosine Related topic PolynomialAnalogonForFermatsLastTheorem Defines Weierstrass substitution Defines Weierstaß substitution Defines universal trigonometric substitution