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tan ⁡ 1 2 ∘ tan ⁡ 4 8 ∘ tan ⁡ 5 4 ∘ tan ⁡ 7 2 ∘ = ? \large \tan 12^\circ \tan 48^\circ \tan 54^\circ \tan 72^\circ = ?


The answer is 1.

Ir J by IR J , 21, Philippines

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1 solution

Chew-Seong Cheong
Aug 26, 2018

P = tan ⁡ 1 2 ∘ tan ⁡ 4 8 ∘ tan ⁡ 5 4 ∘ tan ⁡ 7 2 ∘ Note that tan ⁡ ( 9 0 ∘ − θ ) = cot ⁡ θ = tan ⁡ 1 2 ∘ tan ⁡ ( 6 0 ∘ − 1 2 ∘ ) cot ⁡ 3 6 ∘ tan ⁡ ( 6 0 ∘ + 1 2 ∘ ) and tan ⁡ θ tan ⁡ ( 6 0 ∘ − θ ) tan ⁡ ( 6 0 ∘ + θ ) = tan ⁡ ( 3 θ ) = tan ⁡ 3 6 ∘ cot ⁡ 3 6 ∘ = 1 \begin{aligned} P & = \tan 12^\circ \tan 48^\circ {\color{#3D99F6} \tan 54^\circ} \tan 72^\circ & \small \color{#3D99F6} \text{Note that }\tan (90^\circ - \theta) = \cot \theta \\ & = \color{#D61F06}\tan 12^\circ \tan (60^\circ - 12^\circ){\color{#3D99F6} \cot 36^\circ} \tan (60^\circ + 12^\circ) & \small \color{#D61F06} \text{and }\tan \theta \tan (60^\circ-\theta) \tan (60^\circ+\theta) = \tan (3\theta) \\ & = \color{#D61F06}\tan 36^\circ \color{#3D99F6} \cot 36^\circ \\ & = \boxed 1 \end{aligned}

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