Harmonic oscillator. Rewrite $$\frac{d\theta}{dt}=\omega+A\cos\theta +B\sin\theta, A,B \in \mathbb R$$ as $$\frac{d\theta}{dt}=\omega+r\cos(\theta + \alpha), r>0, \alpha \in [-\pi,\pi) \in \mathbb R.$$ Show that unique $A,B$ define unique $r,\theta.$
I have a feeling that we have to use $\cos(\theta + \alpha) = \cos\theta \cos\alpha - \sin\theta \sin\alpha$, but I don't know how to apply that.
Yes, you may use that formula. Note that by letting $A=r\cos(\alpha)$ and $B=-r\sin(\alpha)$, we have that $$r:=\sqrt{A^2+B^2}.$$ Then there is a unique point along the unit circle $(A/r,-B/r)$ such that $$\begin{cases}\cos(\alpha)=\frac{A}{r}\\ \sin(\alpha)=-\frac{B}{r}\end{cases}$$ with $\alpha \in [-\pi,\pi)$.