Identifying a quotient group (NBHM-$2014$)
I think you can set this map $$f:\mathbb C^*\to U,~~z=a+ib\mapsto \frac{a}{|z|}+i\frac{b}{|z|}$$ wherein $U=\{z\in\mathbb C^*\mid|z|=1\}$. Show this map is surjective with $\ker f=\mathbb R^*_{>0}$.
One another way,
The map $f:\mathbb{C}^* \to \mathbb{C}^*$ defined by $f(z=r.e^{i\theta})=1.e^{i\theta}$ is a homomorphism with kernel being the set of positive reals.
By the Fundamental theorem of homomorphism we can say that $\mathbb{C}^*/P$ is isomorphic to $f(\mathbb{C}^*)=S^1$. where $S^1=\{z\in\mathbb{C} | |z|=1\}$.