establish the identity

[alyren]

establish the identity
[cos(x-y)]/[cos(x+y)] = [1+ tanx tany] / [1- tanx tany]

got stuck in this step.
(cosx cosy + sinx siny) / (cosx cosy - sinx siny)

 

[soroban]

Hello, alyren!

\(\displaystyle \text{Establish the identity: }\;\frac{\cos(x-y)}{\cos(x+y)} \:=\: \frac{1+ \tan x\tan y}{1- \tan x\tan y}\)

\(\displaystyle \text{You had: }\;\frac{\cos(x-y)}{\cos(x+y)} \;\;=\;\;\frac{\cos x\cos y + \sin x\sin y}{cos x\cos y - \sin x\sin y}\)


\(\displaystyle \text{Divide numerator and denominator by }\cos x\cos y\)

. . \(\displaystyle \frac{\dfrac{\cos x\cos y}{\cos x\cos y} + \dfrac{\sin x\sin y}{\cos x\cos y}} {\dfrac{\cos x\cos y}{\cos x\cos y} - \dfrac{\sin x\sin y}{\cos x\cos y}} \;\;=\;\;\frac{1 + \dfrac{\sin x}{\cos x}\!\cdot\!\dfrac{\sin y}{\cos y}}{1 - \dfrac{\sin x}{\cos x}\!\cdot\!\dfrac{\sin y}{\cos y}} \;\;=\;\;\frac{1 + \tan x\tan y}{1 - \tan x\tan y}\)