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 ProofOfDoubleAngleIdentity 2013-03-22 12:50:30 2013-03-22 12:50:30 drini (3) drini (3) 4 drini (3) Proof msc 51-00