Is there a formula for $\sin(xy)$ Can we express a trigonometric function for the product of two angles as a function of trigonometric functions of its factors?
For example: Is there a formula for $\sin(xy)$ as a function of $\sin x$ and $\sin y$ or other trigonometric functions of $x$ and $y$.

Unlikely to be explicitly/purely in terms of $\sin x$, $\sin y$, etc.
But 
$\, \sin xy=\sin \left[ \frac{(x+y)^{2}}{4}-\frac{(x-y)^{2}}{4} \right]
           =\sin \frac{(x+y)^{2}}{4} \cos \frac{(x-y)^{2}}{4}-
            \cos \frac{(x+y)^{2}}{4} \sin \frac{(x-y)^{2}}{4}$
or approximately for small $\, x$ and $y$,
$\sin xy \approx \sin x \sin y
 \left( 1+\frac{\sin^{2} x+\sin^{2} y}{6}+\frac{3}{40}\sin^{4} x-\frac{5}{36}\sin^{2} x \sin^{2} y+\frac{3}{40} \sin^{4} y \right)$
