From my course:

\(\displaystyle (cos(\theta) + jsin(\theta))^3 = cos(3\theta) + jsin(3\theta)\)

\(\displaystyle cos^3(\theta) + 3jcos^2(\theta)sin(\theta) - 3cos(\theta)sin^2(\theta) - jsin^3(\theta) = cos(3\theta) + jsin(3\theta)\)

<->

\(\displaystyle cos(3\theta) = cos^3(\theta) - 3cos(\theta)sin^2(\theta)\) and \(\displaystyle sin(3\theta) = 3cos^2(\theta)sin(\theta) - sin^3(\theta)\)

I have no problem with the first two lines, I don't understand how you can logically deduce the last line from the first two lines.

After all, \(\displaystyle cos(3\theta) = cos^3(\theta) + 3jcos^2(\theta)sin(\theta) - 3cos(\theta)sin^2(\theta) - jsin^3(\theta) - jsin(3\theta)\)