Question

If $\frac{\cos \left(x-y\right)}{\cos \left(x+y\right)}=\frac{m}{n}$, then write the value of tan x tan y.

Answer 1

$$\begin{gathered} \frac{\cos (x-y)}{\cos (x+y)}=\frac{m}{n} \\ \Rightarrow \frac{\cos x\cos y+\sin x\sin y}{\cos x\cos y-\sin x\sin y}=\frac{m}{n} \\ \Rightarrow \frac{1+\tan x\tan y}{1-\tan x\tan y}=\frac{m}{n}\left[\mathrm{Dividing}\mathrm{numerator}\mathrm{and}\mathrm{denominator}\mathrm{of}\text{LHS}\mathrm{by}\cos x\cos y\right] \\ \Rightarrow n+n\tan x\tan y=m-m\tan x\tan y \\ \Rightarrow \tan x\tan y(m+n)=m-n \\ \Rightarrow \tan x\tan y=\frac{m-n}{m+n} \end{gathered}$$