If cos(θ1−θ2)cos(θ1+θ2)+cos(θ3+θ4)cos(θ3−θ4)=0 then |tanθ1tanθ2tanθ3tanθ4|=

If cos(θ1−θ2)cos(θ1+θ2)+cos(θ3+θ4)cos(θ3−θ4)=0 ⇒cos(θ1−θ2)cos(θ1+θ2)=−cos(θ3+θ4)cos(θ3−θ4) ⇒cos(θ1−θ2)+cos(θ1+θ2)cos(θ1−θ2)−cos(θ1+θ2)=−cos(θ3+θ4)+cos(θ3−θ4)−cos(θ3+θ4)−cos(θ3−θ4) ⇒2cosθ1.cosθ22sinθ1.sinθ2=2sinθ3.sinθ4−2cosθ3.cosθ4 ⇒cotθ1.cotθ2=−tanθ3.tanθ4 ⇒tanθ.tanθ2.tanθ3.tanθ4=−1 |tanθ1tanθ2tanθ3tanθ4|=|−1|=1

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If $\frac{\cos (θ_{1} - θ_{2})}{\cos (θ_{1} + θ_{2})} + \frac{\cos (θ_{3} + θ_{4})}{\cos (θ_{3} - θ_{4})} = 0$ then $| \tan θ_{1} \tan θ_{2} \tan θ_{3} \tan θ_{4} | =$

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Detailed Solution

\frac{\cos (θ_{1} - θ_{2})}{\cos (θ_{1} + θ_{2})} + \frac{\cos (θ_{3} + θ_{4})}{\cos (θ_{3} - θ_{4})} = 0

$\Rightarrow \frac{\cos (θ_{1} - θ_{2})}{\cos (θ_{1} + θ_{2})} = - \frac{\cos (θ_{3} + θ_{4})}{\cos (θ_{3} - θ_{4})}$

$\Rightarrow \frac{\cos (θ_{1} - θ_{2}) + \cos (θ_{1} + θ_{2})}{\cos (θ_{1} - θ_{2}) - \cos (θ_{1} + θ_{2})} = \frac{- \cos (θ_{3} + θ_{4}) + \cos (θ_{3} - θ_{4})}{- \cos (θ_{3} + θ_{4}) - \cos (θ_{3} - θ_{4})}$