Something interesting can be said using the first fact. We know that [tex] \sin(x+y) [/tex] gives us some combination of [tex] \cos(x),\cos(y), [/tex] and we know that [tex] \sin(2y) [/tex] gives us [tex] 2\sin(x)\cos(x) [/tex] so the first fact tells us something that will allow us to reduce the second fact nicely:
[tex] \cos{x+y}=\cos{x}\cos{y}-\sin{x}\sin{y} =0\implies\\
\cos{x}\cos{y}=\sin{x}\sin{y} [/tex]
Now, expanding the second part gives:
[tex] \sin{2y}\cos{x}+\cos{2y}\sin{x}=-\sin{x}\implies\\
2\sin{y}\cos{y}*\cos{x}+(\cos^2{y}-\sin^2{y})(\sin{x})=\sin{x}\implies\\
2\sin^2{y}\sin{x}+(1-2\sin^2{y})(\sin{x})=\sin{x}\implies\\
\sin^2{y}\sin{x}+\sin{x}-2\sin^2{y}\sin{x}=\sin{x}\implies\\
\sin{x}=\sin{x} [/tex]
Our original assumptions were true; we were able to use the fact to reduce the second equation to an equality.
Also, I think you mean [tex] \sin{x} [/tex]rather than -[tex] \sin{x} [/tex]
