# double angle identity

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