If \(\sin (y+z-x), \sin (z+x-y)\) and \(\sin (x+y-z)\) are in A.P., then

As \(\sin (y+z-x), \sin (z+x-y)\) and \(\sin (x+y-z)\) are in A.P.
\(\therefore \sin (z+x-y)-\sin (y+z-x)=\sin\)\((x+y-z)-\sin (z+x-y)\)
\(\Rightarrow 2 \cos z \sin (x-y)=2 \cos x \sin (y-z)\)
\(\Rightarrow \cos z \sin (x-y)=\cos x \sin (y-z)\)
\(\Rightarrow \frac{\cos z \sin (x-y)}{\cos x \cos y \cos z}=\frac{\cos x \sin (y-z)}{\cos x \cos y \cos z}\)
\(\Rightarrow \frac{\sin (x-y)}{\cos x \cos y}=\frac{\sin (y-z)}{\cos y \cos z}\)
Using \(\frac{\sin (A-B)}{\cos A \cos B}=\tan A-\tan B\)
\(\Rightarrow \tan x-\tan y=\tan y-\operatorname{taz}\)
\(\Rightarrow \tan y-\tan x=\tan z-\tan y\)
\(\therefore \tan x, \tan y \operatorname{and} \tan z \operatorname{are}\) in A.P.