I have been trying to understand this problem from The Theoretical Minimum. The following solution is given for the equation of motion of a harmonic oscillator. $$ \vec{r}(t)=\vec{c}_1\cos(\omega t)+\vec{c}_2\sin(\omega t) $$ Its rewritten for a given angle $\theta$. $$ \vec{r}(t)=\vec{b}_1\cos(\omega t - \theta)+\vec{b}_2\sin(\omega t - \theta) $$ Where $$ \vec{b}_1=\vec{c}_1\cos(\theta)+\vec{c}_2\sin(\theta) $$ and $$ \vec{b}_2=\vec{c}_2\cos(\theta)-\vec{c}_1\sin(\theta) $$
I want to know what identity or trigonometric property is this. Thanks.
They are exploiting the following identities:
Use the last two on the equation: $$\vec{r}(t)=\vec{b}_1\cos(\omega t - \theta) + \vec{b}_2\sin(\omega t - \theta)$$ expand it and then group identical terms of sine and cosine of $\omega t$, and you should reach the first version independent of $\theta$ (well, is just a constant so it still hiden into the other constants).