Sum of two trig function's identity We all know that
$\sin(x) + \sin(y) = 2\sin((x+y)/2)\cos((x-y)/2)$
But is there an identity for
$\sin(x) + z\sin(y) = ?$
Or do I need to figure it out using Euler's formula
$\sin(x) = (e^{ix} - e^{-ix})/2$ and put it back into trigonometric form?

As far as I know, there is no simple relation turning a sum in a product, except in the case $|z|=1$.
To convince yourself, consider the zero set of the LHS expression:
$$\sin(x)+\sin(y)=0,$$
giving the solutions
$$\sin(y)=-\sin(x),$$
$$y=-x+2k\pi\lor y=x+\pi+2k\pi,$$
$$y+x=2k\pi\lor y-x=(2k+1)\pi.$$
This zero set is formed of two pencils of parallel straight lines, which correspond to the zero sets of two independent sine waves, having for arguments a linear combination of $x$ and $y$.
With $|z|\ne1$, the relation is more complicated:
$$\sin(x)+k\sin(y)=0$$
does not define straight lines but transcendent curves and no linear combination is possible.
