complex numbers: establish the identity??

[Xupica]

I am doing this worksheet on complex numbers and I don't understand how to get to the final identity given in question 2. I have the worked solutions (show in the screenshot) but I don't understand how they simplified it to 2Cos4θ + 8cos2θ +6 ?

 

[Dr.Peterson]

They used the (unstated) fact that [MATH]2\cos(\theta) = e^{i\theta} + e^{-i\theta}[/MATH], which is derived in much the same way as the "result" they give you.

 

[Xupica]

They used the (unstated) fact that [MATH]2\cos(\theta) = e^{i\theta} + e^{-i\theta}[/MATH], which is derived in much the same way as the "result" they give you.

Oooh i see, I completely forgot about that!! thank you