Processing math: 21%

Wednesday, April 29, 2015

Prove tan55tan65tan75=tan85 (Method II)

tan55tan65tan75

=tan55tan65tan75tan85tan85

=tan85tan55tan65tan75tan85

=tan85sin55sin65sin75(cos85)cos55cos65cos75sin85

=\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.)

No comments:

Post a Comment