Sine of Integer Multiple of Argument/Formulation 2/Examples/Sine of Quintuple Angle

Example of Use of Sine of Integer Multiple of Argument: Formulation 2

$\sin 5 \theta = \cos^5 \theta \paren {5 \tan \theta - 10 \tan^3 \theta + \tan^5 \theta}$

$\ds \sin 5 \theta$

$=$

$\ds \cos^n \theta \paren {\dbinom n 1 \tan \theta - \dbinom n 3 \paren {\tan \theta}^3 + \dbinom n 5 \paren {\tan \theta}^5 - \cdots}$

$\ds $

$=$

$\ds \cos^5 \theta \paren {\dbinom 5 1 \tan \theta - \dbinom 5 3 \paren {\tan \theta}^3 + \dbinom 5 5 \paren {\tan \theta}^5 }$

$\ds $

$=$

$\ds \cos^5 \theta \paren {5 \tan \theta - 10 \tan^3 \theta + \tan^5 \theta}$

$\blacksquare$