proof of double angle identity

Sine:

$\displaystyle\sin(2a)$	$\displaystyle=$	$\displaystyle\sin(a+a)$
	$\displaystyle=$	$\displaystyle\sin(a)\cos(a)+\cos(a)\sin(a)$
	$\displaystyle=$	$\displaystyle 2\sin(a)\cos(a).$

Cosine:

$\displaystyle\cos(2a)$	$\displaystyle=$	$\displaystyle\cos(a+a)$
	$\displaystyle=$	$\displaystyle\cos(a)\cos(a)+\sin(a)\sin(a)$
	$\displaystyle=$	$\displaystyle\cos^{2}(a)-\sin^{2}(a).$

By using the identity

\sin^{2}(a)+\cos^{2}(a)=1

we can change the expression above into the alternate forms

\cos(2a)=2\cos^{2}(a)-1=1-2\sin^{2}(a).

$\displaystyle\tan(2a)$	$\displaystyle=$	$\displaystyle\tan(a+a)$
	$\displaystyle=$	$\displaystyle\frac{\tan(a)+\tan(a)}{1-\tan(a)\tan(a)}$
	$\displaystyle=$	$\displaystyle\frac{2\tan(a)}{1-\tan^{2}(a)}.$

Title	proof of double angle identity
Canonical name	ProofOfDoubleAngleIdentity
Date of creation	2013-03-22 12:50:30
Last modified on	2013-03-22 12:50:30
Owner	drini (3)
Last modified by	drini (3)
Numerical id	4
Author	drini (3)
Entry type	Proof
Classification	msc 51-00