help with verifying this problem

[koolaidsman]

I'm horrible with trig identities and I'm stuck on this last problem.
It says to verify using identities

sin 4(theta) = cos(theta) * (4sin(theta) - 8sin^3(theta)

The obvious side to break down is the right side, but I have no clue how to go about doing it.

 

[masters]

I'm horrible with trig identities and I'm stuck on this last problem.
It says to verify using identities

sin 4(theta) = cos(theta) * (4sin(theta) - 8sin^3(theta)

The obvious side to break down is the right side, but I have no clue how to go about doing it.

Hi koolaidsman,

I started from the left and used

[1] $\displaystyle \\ \\ \sin 4\theta=\sin(2(2\theta))$

[2] $\displaystyle \\ \\ \sin 2\theta=2\sin\theta \cos\theta$.

[3] $\displaystyle \\ \\ \cos 2\theta=1-2\sin^2\theta$.

$\displaystyle \sin 4 \theta=$

$\displaystyle \sin (2(2\theta))=$

$\displaystyle 2 \sin 2\theta \cos 2\theta=$

$\displaystyle 2(2 \sin \theta \cos \theta)(1-2\sin^2\theta)=$

$\displaystyle 4 \sin \theta \cos \theta(1-2\sin^2 \theta)=$

$\displaystyle 4 \sin \theta \cos \theta-8 \sin^3 \theta \cos \theta=$

$\displaystyle \boxed{\cos \theta (4 \sin \theta -8 \sin^3 \theta )}$