Show that the set of all complex numbers on the unit circle form a group under multiplication of complex numbers. Some helpful trig identities:
$$\sin (\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \\ \sin (\alpha − \beta) = \sin \alpha \cos \beta − \cos \alpha \sin \beta \\ \cos (\alpha + \beta) = \cos \alpha \cos \beta − \sin \alpha \sin \beta \\ \cos (\alpha − \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta$$
Can some one help me where to start, what I did was I know The set of all points on the unit circle are the complex numbers whose absolute value is $1$ and where $\theta$ is the angle between the positive $x$-axis and the line segment joining the origin and the point, so plugged $1$ into $1+i1= \cos \theta + i \sin \theta$. Then I don’t know what to do next