Given thatsin⁡(y−x)cos⁡(x+y)=12sin⁡(x+y)cos⁡(x−y)=13Determiningsin⁡(2x)As stated in my perspective, the question does not make any sense. We know...

Given that 
${\displaystyle \sin (y-x)\cos (x+y)=\frac{1}{2}}$ 
${\displaystyle \sin (x+y)\cos (x-y)=\frac{1}{3}}$ 
Determining ${\displaystyle \sin \left(2x\right)}$ 
As stated in my perspective, the question does not make any sense. We know that the double angle identity for ${\displaystyle \sin 2x}$ is given by 
${\displaystyle \sin \left(2x\right)=2\sin \cos }$ 
Let us try simpiflying the second equation 
$\sin (x+y)-\cos (x-y)=\sin x\cos y+\cos x\sin y-\cos x\cos y-\sin x\sin y=\cos x(\sin y-\cos y)+\sin x(\cos y-\sin y)=(\cos x-\sin x)(\sin y-\cos y)$