Question

已知tan[（π/4）+α]=2,求（sin2α-cos^2α）/（1+cos2α）的值

Answer 1

tan[（π/4）+α]=2 => (1+tanα)/(1-tanα)=2 =>tanα=1/3
（sin2α-cos²α）/（1+cos2α）
=(2sinαcosα-cos²α)/2cos²α 同除cosα
=(2tanα-1)/2
=-1/6

sin2α=2sinacosα
cos2α=cos²α-sin²α=2cos²α-1=1-sin²α =>cos2α+1=2cos²α
tan（a+b）=（tana+tanb）/（1-tanatanb）

Answer 2

tan(α＋π/4)＝2
(tanα+tanπ/4)/(1-tanαtanπ/4)=2
(tanα+1)/(1-tanα)=2
tanα+1=2-2tanα
tanα=1/3

(sin2α-cos^2α)/(1+cos2α)
= (2sinαcosα-cos^2α)/(1+2cos^2α-1)
= (2sinαcosα-cos^2α)/(2cos^2α)
= tanα-1/2
= 1/3 - 1/2
= -1/6