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

\(\displaystyle \sin 4 \theta=\)

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

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

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

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

\(\displaystyle 4 \sin \theta \cos \theta-8 \sin^3 \theta \cos \theta=\)

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