\(\sin(x-y)=\sin x\cos y-\sin y\cos x\)
Since \(\cos x=\dfrac8{17}\), you have
\(\sin x=\sqrt{1-\cos^2x}=\sqrt{1-\left(\dfrac8{17}\right)^2}=\dfrac{15}{17}\)
and since \(\cos y=\dfrac35\), you have
\(\sin y=\sqrt{1-\cos^2y}=\sqrt{1-\left(\dfrac35\right)^2}=\dfrac45\)
So,
\(\sin(x-y)=\dfrac{15}{17}\times\dfrac35-\dfrac45\times\dfrac8{17}=\dfrac{13}{85}\)
Note that this assumes that both \(\sin x\) and \(\sin y\) are positive, or \(0.
