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Homedouble angle identity

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# 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.

Related:

DeMoivreIdentity, AngleSumIdentity, AdditionFormulasForSineAndCosine

Synonym:

double-angle identity, double angle formula, double-angle formula, double angle formulae, double-angle formulae

Type of Math Object:

Theorem

Major Section:

Reference

## Mathematics Subject Classification

26A09*no label found*33B10

*no label found*

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## Comments

## double angle identity

Also through Euler formula (equivalent to Moivre formula mentioned by Mr. Akrowne):

e^{2i\alpha}=cos{2\alpha}+isin{2\alpha} (1)

(e^{i\alpha})^2=(cos{\alpha}+isin{\alpha})^2=

=(cos{\alpha}^2-sin{\alpha}^2)+i(2sin{\alpha}cos{\alpha}) (2)

and equating.