Winkelfunktionen für weitere Vielfache

$\displaystyle \sin (3x)$	$\displaystyle =$	$\displaystyle 3 \sin x - 4 \sin^3 x$
$\displaystyle \sin (4x)$	$\displaystyle =$	$\displaystyle 8 \sin x \; \cos^3 x - 4 \sin x \; \cos x$
$\displaystyle \sin (5x)$	$\displaystyle =$	$\displaystyle 5 \sin x - 20\sin^3 x + 16 \sin^5 x$
	$\displaystyle =$	$\displaystyle 16 \sin x \; \cos^4 x - 12 \sin x \; \cos^2 x + \sin x$
$\displaystyle \sin (nx)$	$\displaystyle =$	$\displaystyle n \; \sin x \; \cos^{n - 1} x - {n \choose 3} \sin^3 x \; \cos^{n - 3} x + {n \choose 5} \sin^5 x \; \cos^{n - 5} x \; - \; + \; \dots$
	$\displaystyle =$	$\displaystyle \sum_{j=1}^{\lceil\frac{n}{2}\rceil} (-1)^{j+1} {n \choose 2j - 1} \sin^{2j-1} x \; \cos^{n - 2j + 1} x$

$\displaystyle \cos (3x)$	$\displaystyle =$	$\displaystyle 4 \cos^3 x - 3 \cos x$
$\displaystyle \cos (4x)$	$\displaystyle =$	$\displaystyle 8 \cos^4 x - 8 \cos^2 x + 1$
$\displaystyle \cos (5x)$	$\displaystyle =$	$\displaystyle 16 \cos^5 x - 20 \cos^3 x + 5 \cos x$
$\displaystyle \cos (6x)$	$\displaystyle =$	$\displaystyle 32 \cos^6 x - 48 \cos^4 x + 18 \cos^2 x - 1$
$\displaystyle \cos (nx)$	$\displaystyle =$	$\displaystyle \cos^n x - {n \choose 2} \sin^2 x \; \cos^{n - 2} x + {n \choose 4} \sin^4 x \; \cos^{n - 4} x \; - \; + \; \dots$
	$\displaystyle =$	$\displaystyle \; \sum_{j=0}^{\lfloor\frac{n}{2}\rfloor} (-1)^{j} {n \choose 2j} \sin^{2j} x \; \cos^{n - 2j} x$

$\displaystyle \tan (3x)$	$\displaystyle =$	$\displaystyle \frac{ 3 \tan x - \tan^3 x }{ 1 - 3 \tan^2 x }$
$\displaystyle \tan (4x)$	$\displaystyle =$	$\displaystyle \frac{ 4 \tan x - 4 \tan^3 x }{ 1 - 6 \tan^2 x + \tan^4 x }$
$\displaystyle \cot (3x)$	$\displaystyle =$	$\displaystyle \frac{ \cot^3 x - 3 \cot x }{ 3 \cot^2 x - 1 }$
$\displaystyle \cot (4x)$	$\displaystyle =$	$\displaystyle \frac{ \cot^4 x - 6 \cot^2 x + 1 }{ 4 \cot^3 x - 4 \cot x }$