double angle identity

The double angle identities are

$\\displaystyle\\sin(2x)$	$\\displaystyle=$	$\\displaystyle 2\\sin{x}\\cos{x}$	(1)
$\\displaystyle\\cos(2x)$	$\\displaystyle=$	$\\displaystyle\\cos^{2}{x}-\\sin^{2}{x}=2\\cos^{2}{x}-1=1-2\\sin^{2}{x}$	(2)
$\\displaystyle\\tan(2x)$	$\\displaystyle=$	$\\displaystyle\\frac{2\\tan{x}}{1-\\tan^{2}{x}}$	(3)

These are all derived from their respective trigonometric addition formulas. For example,

$\\displaystyle\\sin(2x)$	$\\displaystyle=$	$\\displaystyle\\sin(x+x)$
	$\\displaystyle=$	$\\displaystyle\\sin{x}\\cos{x}+\\cos{x}\\sin{x}$
	$\\displaystyle=$	$\\displaystyle 2\\sin{x}\\cos{x}$

The formula for cosine follows similarly, and the formula tangent is derived by taking the ratio of sine to cosine, as always.

The double angle identities can also be derived from the de Moivre identity.

Title	double angle identity
Canonical name	DoubleAngleIdentity
Date of creation	2013-03-22 12:14:31
Last modified on	2013-03-22 12:14:31
Owner	Wkbj79 (1863)
Last modified by	Wkbj79 (1863)
Numerical id	16
Author	Wkbj79 (1863)
Entry type	Theorem
Classification	msc 26A09
Classification	msc 33B10
Synonym	double-angle identity
Synonym	double angle formula
Synonym	double-angle formula
Synonym	double angle formulae
Synonym	double-angle formulae
Related topic	DeMoivreIdentity
Related topic	AngleSumIdentity
Related topic	AdditionFormulasForSineAndCosine