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

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

$\map \sin {6 \theta } = \cos^6 \theta \paren {6 \tan \theta - 20 \tan^3 \theta + 6 \tan^5 \theta}$

$\ds \map \sin {6 \theta}$

$=$

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

$\ds $

$=$

$\ds \cos^6 \theta \paren {\dbinom 6 1 \tan \theta - \dbinom 6 3 \paren {\tan \theta}^3 + \dbinom 6 5 \paren {\tan \theta}^5 }$

$\ds $

$=$

$\ds \cos^6 \theta \paren {6 \tan \theta - 20 \tan^3 \theta + 6 \tan^5 \theta }$

$\blacksquare$