已知tan(a+π/4)=2 求cos2a-sin2a/2sinacosa+cos方a

题目

已知tan(a+π/4)=2 求cos2a-sin2a/2sinacosa+cos方a

答案

tan(a+π/4)=2
(tana+1)/(1-tana)=2
tana+1=2-2tana
3tana=1
tana=1/3
(cos2a-sin2a)/(2sinacosa+cos^2a)
=(cos^2a-sin^2a-2sinacosa)/(2sinacosa+cos^2a)
上下同除cos^2a
=(1-tan^2a-2tana)/(2tana+1)
=(1-1/9-2/3)/(2/3+1)
=(2/9)/(5/3)
=2/15