Are compositions of rotations a rotation?

Question

The problem asks:
Suppose that $R_\theta$ is the counter-clockwise rotation of any vector in $\mathbb R^2$ by an angle $\theta$ $\epsilon$ $(0,\pi]$. Suppose that $F \circ G$ denotes the composition of the two functions, i.e, $F \circ G(x)$ = $F(G(x))$. Is $R_\theta\circ R_\phi$ a rotation? What is the matrix $M(R_\theta \circ R_\phi)$?
Clearly, it is a rotation but I don't know how to prove how. Any videos or solutions can help. Thank you!

Answer 1

$\pmatrix{\cos\theta&-\sin\theta\\\sin\theta&\cos\theta}\pmatrix{\cos\phi&-\sin\phi\\\sin\phi&\cos\phi}=\pmatrix{\cos(\theta+\phi)&-\sin(\theta+\phi)\\\sin(\theta+\phi)&\cos(\theta+\phi)}$