Question

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

• A. A.P.
• B. G.P.
• C. H.P.
• D. none of these

Answer 1

Answer: (A)

A.P.

As $\sin (y+z-x),\sin (x+z-y),\sin (y+x-z)$
$\Rightarrow \frac{\sin (y+z-x)+\sin (y+x-z)}{2}=\sin (x+z-y)$
$\Rightarrow 2\sin y\cos (z-x)=2\sin (x+z-y)\Rightarrow \sin y\cos (z-x)=-\sin y\cos (z+x)+\sin (z+y)\cos y\Rightarrow \sin y\left(\cos (z-x)+\cos (z+x)\right)=\sin (z+y)\cos y\Rightarrow \tan x+\tan z=2\tan y$