Answer
$\dfrac{\sin(x+y)+\sin (x-y)}{\cos(x+y)+\cos (x-y)}=\tan x$
Work Step by Step
Need to verify $\dfrac{\sin(x+y)+\sin (x-y)}{\cos(x+y)+\cos (x-y)}=\tan x$
This implies that
$\dfrac{\sin x \cos y+\sin y \cos x+\sin x\cos y-\sin y \cos x}{\cos x \cos y -\sin x\sin y +\cos x \cos y+\sin x \sin y}=\dfrac{2 \sin x \cos y}{2 \cos x \cos y}$
or, $\dfrac{\sin x}{\cos x}=\tan x$ (RHS)
Hence, the left-hand side and right hand side are equal.
