calculating cos(nθ) and sin(nθ) with de Moivre

I'm not sure in which subforum I should post this, feel free to move it to another subforum.

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