Answer
$\dfrac {\sin \left( x+y\right) -\sin \left( x-y\right) }{\cos \left( x+y\right) +\cos \left( x-y\right) }=\dfrac {\left( \sin x\cos y+\cos x\sin y\right) -\left( \sin x\cos y-\cos x\sin y\right) }{\left( \cos x\cos {y}-\sin x\sin y\right) +\left( \cos x\cos y+\sin x\sin y\right) }=\dfrac {2\cos x\sin y}{2\cos x\cos y}=\tan y $
Work Step by Step
$\dfrac {\sin \left( x+y\right) -\sin \left( x-y\right) }{\cos \left( x+y\right) +\cos \left( x-y\right) }=\dfrac {\left( \sin x\cos y+\cos x\sin y\right) -\left( \sin x\cos y-\cos x\sin y\right) }{\left( \cos x\cos {y}-\sin x\sin y\right) +\left( \cos x\cos y+\sin x\sin y\right) }=\dfrac {2\cos x\sin y}{2\cos x\cos y}=\tan y $
