In Physics, the Simple Harmonic Oscillator is represented by the equation $d^2x/dt^2=-\omega^2x$ .
By using the characteristic polynomial, you get solutions of the form $x(t)=Ae^{i\omega t} + Be^{-i\omega t}$. I get that you use Euler's formula $e^{i\theta}=\cos\theta + i\sin\theta$, but I can't seem to find my way all the way to the 'traditional form' of $D\cos\omega t + C\sin\omega t$.
I'm stuck here: $A(\cos\omega t + i\sin\omega t) + B(\cos\omega t - i\sin\omega t)$. What am I missing in taking it all the way? It doesn't seem valid to me (not sure why or why not) to make $C = iA - iB$.