The functions $\cos\omega t$ and $\sin\omega t$ each have period $\dfrac{2\pi}{\omega}$. Therefore their linear combination $A\cos\omega t + B\sin\omega t$ also has period $\dfrac{2\pi}{\omega}$. This can be written as $R\cos(\omega t - \phi)$ where $R = \sqrt{A^2+B^2}$ and $\tan\phi = B/A$, confirming the period is $\dfrac{2\pi}{\omega}$.