# How do you express sin(2x)+sin(4x) in terms of sin(x) and cos(x) ?

Nov 17, 2017

In terms of $\sin \left(x\right)$ and $\cos \left(x\right)$ we find:

$\sin \left(2 x\right) + \sin \left(4 x\right) = 2 \sin \left(x\right) \cos \left(x\right) \left(1 + 2 {\cos}^{2} \left(x\right) - 2 {\sin}^{2} \left(x\right)\right)$

#### Explanation:

Note that:

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

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

So:

$\sin \left(4 x\right) = \sin \left(2 \left(2 x\right)\right)$

color(white)(sin(4x)) = 2 sin(2x)(cos(2x)

$\textcolor{w h i t e}{\sin \left(4 x\right)} = 2 \left(2 \sin \left(x\right) \cos \left(x\right)\right) \left({\cos}^{2} \left(x\right) - {\sin}^{2} \left(x\right)\right)$

$\textcolor{w h i t e}{\sin \left(4 x\right)} = 4 \sin \left(x\right) {\cos}^{3} \left(x\right) - 4 {\sin}^{3} \left(x\right) \cos \left(x\right)$

So:

$\sin \left(2 x\right) + \sin \left(4 x\right) = 2 \sin \left(x\right) \cos \left(x\right) + 4 \sin \left(x\right) {\cos}^{3} \left(x\right) - 4 {\sin}^{3} \left(x\right) \cos \left(x\right)$

$\textcolor{w h i t e}{\sin \left(2 x\right) + \sin \left(4 x\right)} = 2 \sin \left(x\right) \cos \left(x\right) \left(1 + 2 {\cos}^{2} \left(x\right) - 2 {\sin}^{2} \left(x\right)\right)$