Is it possible to rewrite $\sin(x) / \sin(y)$ in the form of $\sin(z)$?

I'm looking to get a particular answer in the form of $\sin(z)$, and I managed to reach an answer in the form $\sin(x)/\sin(y)$. I've checked on a calculator which has confirmed that they're the same number, but how can I convert the fraction into a single sin in order to show that without relying on the calculator?

There is no general solution $\sin(x)/\sin(y) = \sin(z)$ (for all $x,y$). This can easily be seen by setting $y=0$ and then the fraction diverges - which $\sin(z)$ would never do.

You will always have $-1\leq\sin z\leq 1$, while $\sin x/\sin y $ can be any real number or even be undefined when $y=0$. For example, $$\frac {\sin\pi/2}{\sin\pi/4}=\frac1 {1/\sqrt2}=\sqrt2>1$$ and no choice of $z $ will give you $\sin z=\sqrt2$.

In the case in which the quotient is between $-1$ and $1$, you can write $$z=\arcsin\left (\frac {\sin x}{\sin y}\right), $$ but I don't think there is a simpler expression.