Explain how solution got $c_1$ and $c_2$

Question

Can someone explain how the solution manual got $c_1$ and $c_2$ in this:

Answer 1

Hint

Notice that it's a linear system with two unknown $c_1$ and $c_2$.

$$c_1\cos\omega t_0+c_2\sin\omega t_0=x_0\implies c_1=\frac{x_0-c_2\sin\omega t_0}{\cos\omega t_0}$$

Then you replace $c_1$ by $\frac{x_0-c_2\sin\omega t_0}{\cos\omega t_0}$ in the second equation to get $c_2$, and you remplace the $c_2$ that you found in the second equation in this equation to get the result.

Answer 2

Note that $$\cos^2A+\sin^2A=1$$ and thus multiply the first equation with $w\cos wt$ and the second with $\sin wt$ and subtract. And vice versa.