Question

请问为什么(1-sinx)/cosx = tan(x/2)?

Answer 1

错了。
(1 -sinx)/cosx = [ 1 -tan (x/2) ] / [ 1 +tan (x/2) ].
应该是：
(1 -cos x) /sin x = tan (x/2),
或：
sin x /(1 +cos x) = tan (x/2).

= = = = = = = = =

1. (1 -cos x) /sin x = tan (x/2).
证明：因为 cos x =1 -2 [ sin (x/2) ]^2,
sin x = 2 sin (x/2) cos (x/2),
所以 (1 -cos x) /sin x = 2 [ sin (x/2) ]^2 / [ 2 sin (x/2) cos (x/2) ]
= sin (x/2) / [ cos (x/2) ]
= tan (x/2).

= = = = = = = = =

2. sin x /(1 +cos x) = tan (x/2).
证明：因为 sin x = 2 sin (x/2) cos (x/2),
cos x = 2 [ cos (x/2) ]^2 -1,
所以 sin x /(1 +cos x) = ... =tan (x/2).

= = = = = = = = =
同理,
(1 +cos x) /sin x =sin x /(1 -cos x) = cot (x/2).

cos x = [ cos (x/2) ]^2 -[ sin (x/2) ]^2
= 1 -2 [ sin (x/2) ]^2
= 2 [ cos (x/2) ]^2 -1,
公式的选取很关键.

Answer 2

1-sinx=sinx/2sinx/2+cosx/2cosx/2-2sinx/2cosx/2
cosx=cosx/2cosx/2-sinx/2sinx/2
分子分母同除以cosx/2cosx/2，得到
(tanx/2tanx/2+1-2tanx/2)/(1-tanx/2)=1-tanx/2的绝对值