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Solve tanθt...

Question

Solve $tan θ tan 4 θ = 1$ for $0 < θ < π$

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Solution

Given, $tan θ tan 4 θ = 1 \Rightarrow sin θ sin 4 θ = cos θ cos 4 θ \Rightarrow cos θ cos 4 θ - sin θ sin 4 θ = 0 \Rightarrow cos (θ + 4 θ) = 0 \Rightarrow cos 5 θ = 0$
$\Rightarrow 5 θ = (2 n + 1) \frac{π}{2} \Rightarrow θ = (2 n + 1) \frac{π}{10}$

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