Question

What is the value of $\frac{\sin (x-y)}{\cos x\cos y}+\frac{\sin (y-z)}{\cos y\cos z}+\frac{\sin (z-x)}{\cos z\cos x}$?

Answer 1

$\frac{\sin (x-y)}{\cos x\cos y}=\frac{\sin x\cos y-\cos x\sin y}{\cos x\cos y}$
$=\frac{\sin x\cos y}{\cos x\cos y}-\frac{\cos x\sin y}{\cos x\cos y}$
$=\tan x-\tan y$
$\therefore$ The given expression
$=\tan x-\tan y+\tan y-\tan z+\tan z+\tan z-\tan x=0$