# What is sin(4x) in terms of sin(2x) and cos(2x)?

## What identities is used to get the answer?

May 30, 2017

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

#### Explanation:

We have: $\sin \left(4 x\right)$

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

Let's apply the angle sum identity for $\sin \left(x\right)$; $\sin \left(\alpha + \beta\right) = \sin \left(\alpha\right) \cos \left(\beta\right) + \cos \left(\alpha\right) \sin \left(\beta\right)$:

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

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

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