三角函數倍角半角

1)
(cos π/7) (cos 2π/7) (cos 4π/7)
= [(1 / (2sin π/7)] * (2sin π/7) (cos π/7) (cos 2π/7) (cos 4π/7)
= [(1 / (2sin π/7)] * (sin 2π/7) (cos 2π/7) (cos 4π/7)
= [(1 / (2sin π/7)] * (1/2) * 2(sin 2π/7) (cos 2π/7) * (cos 4π/7)
= [(1 / (2sin π/7)] * (1/2) * sin(4π/7) (cos 4π/7)
= [(1 / (2sin π/7)] * (1/2) * (1/2) * 2sin(4π/7) (cos 4π/7)
= [(1 / (2sin π/7)] * (1/2) * (1/2) * sin8π/7
= (1/8) * (sin8π/7) / (sin π/7)
= - 1/8
2)
(cosx)^2 - (cosy)^2
= (cosx + cosy)(cosx - cosy)
= 2 cos [(x+y)/2] cos [(x-y)/2] * (- 2) sin [(x+y)/2] sin [(x-y)/2]
= 2 [cos (π/6)] cos [(x-y)/2] * (- 2) [sin (π/6)] sin [(x-y)/2]
= - cos [(x-y)/2] * sin [(x-y)/2]
= - (1/2) 2sin(x-y)
= - sin(x-y)
最大值 = 1
最小值 = - 1

2010-06-16 12:19:59 補充：
更正 :

= - cos [(x-y)/2] * sin [(x-y)/2]

= - (1/2) 2 cos [(x-y)/2] * sin [(x-y)/2]

= - (1/2) sin(x - y)

最大值 = 1/2
最小值 = -1/2

2010-06-16 12:33:28 補充：
對不起又錯了!! 重做 :

2)

(cosx)^2 - (cosy)^2

= (cosx + cosy)(cosx - cosy)

= 2 cos [(x+y)/2] cos [(x-y)/2] * (- 2) sin [(x+y)/2] sin [(x-y)/2]

= 2 [cos (π/6)] cos [(x-y)/2] * (- 2) [sin (π/6)] sin [(x-y)/2]

= √3 cos [(x-y)/2] * {- sin [(x-y)/2]}

= (- √3) (1/2) 2 sin(x-y)/2 cos(x-y)/2

= (- √3 /2) sin(x-y)

2010-06-16 12:33:34 補充：
最大值 = √3 /2
最小值 = - √3 /2