Analytic function on $D$ such that $| f ( z ) | = | \sin z | , \forall z \in$ $D ,$ then prove that $f(z)=c\sin z$ for some constant c

Question

Let $D$ be a domain and let $f ( z )$ be an analytic function on $D$ such that $| f ( z ) | = | \sin z | , \forall z \in$ $D ,$ then prove that $f(z)=c\sin z$ for some constant c.

Can we consider the quotient $\frac{f(z)}{\sin z}$? But I am not getting how the quotient has removable singularity at zeros of $\sin z$

Answer 1

Hint If $\sin(a)=0$ for some $a \in D$ deduce that $f(a)=0$ in $D$ and hence there exists an alaytic function $h(z)$ such that $f(z)=(z-a)h(z)$.

Use this to show that $\frac{f(z)}{\sin(z)}$ has a removable singularity at $z=a$.