Functional square root of sine is not unique, and it cannot be defined in the whole complex plane (according to a theorem of I. N. Baker). You can define it on $(0,\pi)$ with $f(0)=0$, for example. The resulting function is analytic in the component of Fatou set of sine adjacent to zero on the right (see, for example the first figure here). But this function $f$ has no zeros in its domain, therefore a representation as infinite product does not make much sense. I mean that $\log f$ can be defined and any breaking of this log into summands gives you an infinite product representation. Of course this answer is related to a true product, it does not exclude that there is some "formal" product representing this square root, whateven a "formal product" may mean.

Functional square root of sine is not unique, and it cannot be defined in the whole complex plane (according to a theorem of I. N. Baker). You can define it on $(0,\pi)$ with $f(0)=0$, for example. The resulting function is analytic in the component of Fatou set of sine adjacent to zero on the right (see, for example the first figure here). But this function $f$ has no zeros in its domain, therefore a representation as infinite product does not make much sense. I mean that $\log f$ can be defined and any breaking of this log into summands gives you an infinite product representation. Of course this answer is related to a true product, it does not exclude that there is some "formal" product representing this square root, whatever a "formal product" may mean.

Functional square root of sine is not unique, and it cannot be defined in the whole complex plane (according to a theorem of I. N. Baker). You can define it on $(0,\pi)$ with $f(0)=0$, for example. The resulting function is analytic in the component of Fatou set of sine adjacent to zero on the right (see, for example the first figure here). But this function $f$ has no zeros in its domain, therefore a representation as infinite product does not make much sense. I mean that $\log f$ can be defined any any breaking of this log into summands gives you an infinite product representation.

Functional square root of sine is not unique, and it cannot be defined in the whole complex plane (according to a theorem of I. N. Baker). You can define it on $(0,\pi)$ with $f(0)=0$, for example. The resulting function is analytic in the component of Fatou set of sine adjacent to zero on the right (see, for example the first figure here). But this function $f$ has no zeros in its domain, therefore a representation as infinite product does not make much sense. I mean that $\log f$ can be defined and any breaking of this log into summands gives you an infinite product representation. Of course this answer is related to a true product, it does not exclude that there is some "formal" product representing this square root, whateven a "formal product" may mean.