tan55∘tan65∘tan75∘
=tan55∘tan65∘tan75∘⋅tan85∘tan85∘
=tan85∘⋅tan55∘tan65∘tan75∘tan85∘
=tan85∘⋅sin55∘sin65∘sin75∘(cos85∘)cos55∘cos65∘cos75∘sin85∘
=\tan85^{\circ}\cdot \dfrac{(\sin 55^{\circ}\sin 65^{\circ})(\sin 75^{\circ}\cos85^{\circ})}{(\cos 55^{\circ}\cos 65^{\circ})(\cos 75^{\circ}\sin85^{\circ})}
=\tan85^{\circ}\cdot\dfrac{{-\dfrac{1}{2}(\cos 120^{\circ}-\cos 10^{\circ})\frac{1}{2}(\sin160^{\circ}-\sin10^{\circ})}}{{\dfrac{1}{2}(\cos 120^{\circ}+\cos 10^{\circ})\dfrac{1}{2}(\sin160^{\circ}+\sin10^{\circ})}}
=-\tan85^{\circ}\cdot \dfrac{(\cos 120^{\circ}-\cos 10^{\circ})(\sin160^{\circ}-\sin10^{\circ})}{(\cos 120^{\circ}+\cos 10^{\circ})(\sin160^{\circ}+\sin10^{\circ})}
=-\tan85^{\circ}\cdot \dfrac{-\dfrac{1}{2}\sin160^{\circ}+\frac{1}{2}\sin10^{\circ}-\sin160^{\circ}\cos 10^{\circ}+\sin10^{\circ}\cos10^{\circ}}{-\frac{1}{2}\sin160^{\circ}-\frac{1}{2}\sin10^{\circ}+\sin160^{\circ}\cos 10^{\circ}+\sin10^{\circ}\cos10^{\circ}}
=-\tan85^{\circ}\cdot \dfrac{-\frac{1}{2}\sin20^{\circ}+\dfrac{1}{2}\sin10^{\circ}-\sin160^{\circ}\cos 10^{\circ}+\dfrac{1}{2}\sin20^{\circ}}{-\dfrac{1}{2}\sin20^{\circ}-\dfrac{1}{2}\sin10^{\circ}+\sin160^{\circ}\cos 10^{\circ}+\dfrac{1}{2}\sin20^{\circ}}
=-\tan85^{\circ}\cdot \dfrac{\dfrac{1}{2}\sin10^{\circ}-\sin160^{\circ}\cos 10^{\circ}}{-\frac{1}{2}\sin10^{\circ}+\sin160^{\circ}\cos 10^{\circ}}
=-\tan85^{\circ}\cdot (-1)
=\tan85^{\circ} (Q.E.D.)
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