正切半角式tan^2{x/2}=(1-cosx)/(1+cosx)tan(x/2)=sinx/(1+cosx)=(1-cosx)/sinx怎么推导出来的啊?谁帮我

Question

正切半角式
tan^2{x/2}=(1-cosx)/(1+cosx) tan(x/2)=sinx/(1+cosx)=(1-cosx)/sinx 怎么推导出来的啊? 谁帮我

Answer 1

利用2倍角的正、余弦公式：
sinx/(1+cosx)= 2sin(x/2)cos(x/2) / [ 2cos²(x/2)]
= sin(x/2) / cos(x/2)
= tan(x/2) …… ①
(1-cosx)/sinx = 2sin²(x/2) / [ 2sin(x/2)cos(x/2) ]
= sin(x/2) / cos(x/2)
= tan(x/2) ……②
由 ①、②得：tan(x/2)=sinx/(1+cosx)=(1-cosx)/sinx
(1-cosx)/(1+cosx) = 2sin²(x/2) / 2cos²(x/2)
= tan²(x/2)
the end