# How do you prove that sin5x=sinx(cos^2 2x-sin^2 2x)+2cosxcos2xsin2x?

Oct 22, 2016

• $\sin \left(\alpha + \beta\right) = \sin \left(\alpha\right) \cos \left(\beta\right) + \cos \left(\alpha\right) \sin \left(\beta\right)$

along with the double angle identities:

• $\sin \left(2 \theta\right) = 2 \sin \left(\theta\right) \cos \left(\theta\right)$
• $\cos \left(2 \theta\right) = {\cos}^{2} \left(\theta\right) - {\sin}^{2} \left(\theta\right)$

we have

$\sin \left(5 x\right) = \sin \left(x + 4 x\right)$

$= \sin \left(x\right) \cos \left(4 x\right) + \cos \left(x\right) \sin \left(4 x\right)$

$= \sin \left(x\right) \cos \left(2 \cdot 2 x\right) + \cos \left(x\right) \sin \left(2 \cdot 2 x\right)$

$= \sin \left(x\right) \left({\cos}^{2} \left(2 x\right) - {\sin}^{2} \left(2 x\right)\right) + \cos \left(x\right) \left(2 \sin \left(2 x\right) \cos \left(2 x\right)\right)$

$= \sin \left(x\right) \left({\cos}^{2} \left(2 x\right) - {\sin}^{2} \left(2 x\right)\right) + 2 \cos \left(2 x\right) \cos \left(2 x\right) \sin \left(2 x\right)$