已知tan(π/4+α)=3,求sin2α - 2(cosα)^2的值

题目

已知tan(π/4+α)=3,求sin2α - 2(cosα)^2的值

答案

tan(π/4+α)=3
(tanπ/4+tana)/(1-tanπ/4tana)=3
(1+tana)/(1-tana)=3
1+tana=3-3tana
tana=2/4=1/2
sin2a=2tana/(1+tan^2a)
=2*1/2 /(1+(1/2)^2)
=4/5
2(cosa)^2=cos2a+1
=(1-tan^2a)/(1+tan^2a) +1
=(1-(1/2)^2)/(1+(1/2)^2)+1
=3/5+1
=8/5
sin2a-2(cosα)^2
=4/5-8/5
=-4/5