# 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 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