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Question

$If \frac{3 - \tan^{2} \frac{π}{7}}{1 - \tan^{2} \frac{π}{7}} = λcos \frac{π}{7}, then λ is$ ________.

Moderate

Solution

$\begin{array}{l} Let θ = \frac{π}{7} \Rightarrow 3 θ = π - 4 θ \\ \Rightarrow \sin 3 θ = \sin 4 θ \\ \Rightarrow 3 \sin θ - 4 \sin^{3} θ = 4 \sin θcos θcos 2 θ \\ \Rightarrow 3 - 4 \sin^{2} θ = 4 \cos θ (2 \cos^{2} θ - 1) \end{array}$
$\begin{array}{l} \Rightarrow 4 \cos θ (2 \cos^{2} θ - 1) = 4 \cos^{2} θ - 1 \\ \Rightarrow 4 \cos θ = \frac{4 \cos^{2} θ - 1}{2 \cos^{2} θ - 1} = \frac{3 - \tan^{2} θ}{1 - \tan^{2} θ} \\ \Rightarrow 4 \cos \frac{π}{7} = \frac{3 - \tan^{2} \frac{π}{7}}{1 - \tan^{2} \frac{π}{7}} \end{array}$

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