If cos(x − y)/cos(x + y) = m/n, Find tan x tan y

Solution

Using, \[ \cos(x-y)=\cos x\cos y+\sin x\sin y \]

and \[ \cos(x+y)=\cos x\cos y-\sin x\sin y \]

\[ \frac{ \cos x\cos y+\sin x\sin y }{ \cos x\cos y-\sin x\sin y } = \frac{m}{n} \]

Dividing numerator and denominator by \[ \cos x\cos y \]

\[ \frac{1+\tan x\tan y}{1-\tan x\tan y} = \frac{m}{n} \]

Let \[ \tan x\tan y=t \]

\[ \frac{1+t}{1-t}=\frac{m}{n} \]

\[ n(1+t)=m(1-t) \]

\[ n+nt=m-mt \]

\[ t(n+m)=m-n \]

\[ \boxed{ \tan x\tan y=\frac{m-n}{m+n} } \]