Cos20 градусов*cos40градусов*cos60градусов*cos80градусов=1/16. Докажите тождество

Question

Cos20 градусов*cos40градусов*cos60градусов*cos80градусов=1/16. Докажите тождество

Answer 1

сos20°·cos40°·cos60°·cos80° = 1/16

сos20°·cos40°·cos60°·cos80° = 1/16

cos60° = 1/2

1/2 · (сos20°·cos40°·cos80°) = 1/16

1/2 · (сos20°·cos40°)·cos80° = 1/16

1/2 · 1/2 ·(сos60° + cos20°)·cos80° = 1/16

1/4 (сos60° + cos20°)·cos80° = 1/16

1/4 · (сos60°·cos80° + cos20°·cos80°) = 1/16

1/4 · (1/2 ·cos80° + 1/2 ·(cos60° + cos100°) = 1/16

1/8 · (cos80° + cos60° + cos100°) = 1/16

1/8 · (cos (90° - 10°) +1/2 + cos(90°+10°)) = 1/16

1/8 · (sin10° +1/2 - sin10°) = 1/16

1/8 · 1/2  = 1/16

1/16≡ 1/16

тождество доказано

Answer 2

умножим и разделим на sin20 градусов

получим:

(2*1/2*sin20*cos20*cos40*cos60*cos80)/sin20=1/16

(1/2*1/2*2*sin40*cos40*cos60*cos80)/sin20=1/16

(1/4sin80*cos60*cos80)/sin20=1/16

(1/8*sin160*cos60)/sin20=1/16

(1/8*sin20*1/2)/sin20=1/16

1/16=1/16