Rewrite $y = \cos(t) - \sin(t)$ to $y = A\cos(\omega t - \phi)$

Question

I'm really lost here.. The subject is differential equations, but I seemed to have missed this part in class. How do I convert to only $\cos$?

$$y = \cos(t) - \sin(t) \rightarrow y = A\cos(\omega t - \phi)$$

Answer 1

$$A\cos(\omega t-\phi)=A\cos\omega t\cos\phi+A\sin\omega t\sin\phi.$$

Then by identification when $\omega=1$,

$$ A\cos\phi=1,\\ A\sin\phi=-1.$$

Solve for $A$ and $\phi$. (Hint: take the sum of squares and take the ratio.)

Answer 2

use $$\cos(t)-\sin(t)=\sqrt{2}\sin\left(\frac{\pi}{4}-t\right)$$