Can Euler's Formula Simplify Trigonometric Proofs?

[mathhelp123]

Hi. I am trying to prove that (2sin^4x(2cos^2x+1))/3 = [sin^2(2x)/6-1/3][cos(2x)].

I tried fixing the right side, changing cos2x to 1 - 2sin^2x, and I went all the way to

(4sin^2x-4sin^4x-2-8sin^4x+8sin^6x+4sin^2x)/6.

I am clueless on how to continue the proof. Please help. Thanks.

 

[Tide]

Your notation isn't very clear but I think this may help:

$$e^{4 i \theta} = \left( e^{i \theta} \right)^4$$

and apply Euler's formula to the individual exponentials.