已知锐角α满足cosα-sinα=√5/5,则(sin2α-cos2α+1)/(1-tanα)=

题目

已知锐角α满足cosα-sinα=√5/5,则(sin2α-cos2α+1)/(1-tanα)=

答案

cosα-sinα=√5/5
1-2sinacosa=1/5
2sinacosa=4/5
1+2sinacosa=9/5
cosa+sina=3√5/5 (a为锐角,负值已舍)
cosa=2√5/5
sina=√5/5
tana=1/2
sin2a=2sinacosa=2 ×2√5/5× √5/5=4/5
cos2a= cos²a-sin²a=4/5-1/5=3/5
分别代入
(4/5-3/5+1)/(1-1/2)=6/5/1/2=12/5