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