Question

Sin(x-y) if cos x =8/17 and cos y = 3/5

Answer 1

$\sin(x-y)=\sin x\cos y-\sin y\cos x$

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 $0$.