tan55∘tan65∘tan75∘
=tan55∘tan65∘tan75∘⋅tan85∘tan85∘
=tan85∘⋅tan55∘tan65∘tan75∘tan85∘
=tan85∘⋅sin55∘sin65∘sin75∘(cos85∘)cos55∘cos65∘cos75∘sin85∘
=tan85∘⋅(sin55∘sin65∘)(sin75∘cos85∘)(cos55∘cos65∘)(cos75∘sin85∘)
=tan85∘⋅−12(cos120∘−cos10∘)12(sin160∘−sin10∘)12(cos120∘+cos10∘)12(sin160∘+sin10∘)
=−tan85∘⋅(cos120∘−cos10∘)(sin160∘−sin10∘)(cos120∘+cos10∘)(sin160∘+sin10∘)
=−tan85∘⋅−12sin160∘+12sin10∘−sin160∘cos10∘+sin10∘cos10∘−12sin160∘−12sin10∘+sin160∘cos10∘+sin10∘cos10∘
=-\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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