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Wednesday, April 29, 2015

Prove tan55tan65tan75=tan85 (Method II)

tan55tan65tan75

=tan55tan65tan75tan85tan85

=tan85tan55tan65tan75tan85

=tan85sin55sin65sin75(cos85)cos55cos65cos75sin85

=tan85(sin55sin65)(sin75cos85)(cos55cos65)(cos75sin85)

=tan8512(cos120cos10)12(sin160sin10)12(cos120+cos10)12(sin160+sin10)

=tan85(cos120cos10)(sin160sin10)(cos120+cos10)(sin160+sin10)

=tan8512sin160+12sin10sin160cos10+sin10cos1012sin16012sin10+sin160cos10+sin10cos10

=-\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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