Consider $z \in \mathbb{C}$ with $|z|=1$ and $\arg z=\theta$, where $\theta \in[0, \pi)$.

(a) Prove algebraically that the modulus of $1+z$ is $2 \cos \frac{1}{2} \theta$ and that the argument is $\frac{1}{2} \theta$. Obtain these results geometrically using the Argand diagram.

(b) Obtain corresponding results algebraically and geometrically for $1-z$.